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Theorem tz7.49 7405
Description: Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)
Hypotheses
Ref Expression
tz7.49.1 𝐹 Fn On
tz7.49.2 (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
Assertion
Ref Expression
tz7.49 ((𝐴𝐵𝜑) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem tz7.49
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ne 2781 . . . . . . . . 9 ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹𝑥)) = ∅)
21ralbii 2962 . . . . . . . 8 (∀𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) ≠ ∅ ↔ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅)
3 tz7.49.2 . . . . . . . . 9 (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
4 ralim 2931 . . . . . . . . 9 (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (∀𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
53, 4sylbi 205 . . . . . . . 8 (𝜑 → (∀𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
62, 5syl5bir 231 . . . . . . 7 (𝜑 → (∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅ → ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
7 tz7.49.1 . . . . . . . . 9 𝐹 Fn On
87tz7.48-3 7404 . . . . . . . 8 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ 𝐴 ∈ V)
9 elex 3184 . . . . . . . 8 (𝐴𝐵𝐴 ∈ V)
108, 9nsyl3 131 . . . . . . 7 (𝐴𝐵 → ¬ ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
116, 10nsyli 153 . . . . . 6 (𝜑 → (𝐴𝐵 → ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅))
12 dfrex2 2978 . . . . . 6 (∃𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) = ∅ ↔ ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅)
1311, 12syl6ibr 240 . . . . 5 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) = ∅))
14 imaeq2 5368 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1514difeq2d 3689 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 ∖ (𝐹𝑥)) = (𝐴 ∖ (𝐹𝑦)))
1615eqeq1d 2611 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹𝑥)) = ∅ ↔ (𝐴 ∖ (𝐹𝑦)) = ∅))
1716onminex 6877 . . . . 5 (∃𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) = ∅ → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅))
1813, 17syl6 34 . . . 4 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅)))
19 df-ne 2781 . . . . . . 7 ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹𝑦)) = ∅)
2019ralbii 2962 . . . . . 6 (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ↔ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅)
2120anbi2i 725 . . . . 5 (((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ↔ ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅))
2221rexbii 3022 . . . 4 (∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ↔ ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅))
2318, 22syl6ibr 240 . . 3 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅)))
24 nfra1 2924 . . . . 5 𝑥𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
253, 24nfxfr 1770 . . . 4 𝑥𝜑
26 simpllr 794 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅)
27 fnfun 5888 . . . . . . . . . . . . . . . . 17 (𝐹 Fn On → Fun 𝐹)
287, 27ax-mp 5 . . . . . . . . . . . . . . . 16 Fun 𝐹
29 fvelima 6143 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ∈ (𝐹𝑥)) → ∃𝑦𝑥 (𝐹𝑦) = 𝑧)
3028, 29mpan 701 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝐹𝑥) → ∃𝑦𝑥 (𝐹𝑦) = 𝑧)
31 nfv 1829 . . . . . . . . . . . . . . . . 17 𝑦𝜑
32 nfra1 2924 . . . . . . . . . . . . . . . . 17 𝑦𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅
3331, 32nfan 1815 . . . . . . . . . . . . . . . 16 𝑦(𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅)
34 nfv 1829 . . . . . . . . . . . . . . . 16 𝑦(𝑥 ∈ On → 𝑧𝐴)
35 rsp 2912 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝑦𝑥 → (𝐴 ∖ (𝐹𝑦)) ≠ ∅))
3635adantld 481 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝐴 ∖ (𝐹𝑦)) ≠ ∅))
37 onelon 5651 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3815neeq1d 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑦)) ≠ ∅))
39 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
4039, 15eleq12d 2681 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) ↔ (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦))))
4138, 40imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦 → (((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) ↔ ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4241rspcv 3277 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ On → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
433, 42syl5bi 230 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (𝜑 → ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4443com23 83 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ On → ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝜑 → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4537, 44syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝜑 → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4636, 45sylcom 30 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝜑 → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4746com3r 84 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4847imp 443 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦))))
4948expcomd 452 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑦𝑥 → (𝑥 ∈ On → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
50 eldifi 3693 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → (𝐹𝑦) ∈ 𝐴)
51 eleq1 2675 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ 𝐴𝑧𝐴))
5250, 51syl5ibcom 233 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → ((𝐹𝑦) = 𝑧𝑧𝐴))
5349, 52syl8 73 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑦𝑥 → (𝑥 ∈ On → ((𝐹𝑦) = 𝑧𝑧𝐴))))
5453com34 88 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑦𝑥 → ((𝐹𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧𝐴))))
5533, 34, 54rexlimd 3007 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (∃𝑦𝑥 (𝐹𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧𝐴)))
5630, 55syl5 33 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑧 ∈ (𝐹𝑥) → (𝑥 ∈ On → 𝑧𝐴)))
5756com23 83 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑥 ∈ On → (𝑧 ∈ (𝐹𝑥) → 𝑧𝐴)))
5857imp 443 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝑧 ∈ (𝐹𝑥) → 𝑧𝐴))
5958ssrdv 3573 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝐹𝑥) ⊆ 𝐴)
60 ssdif0 3895 . . . . . . . . . . . 12 (𝐴 ⊆ (𝐹𝑥) ↔ (𝐴 ∖ (𝐹𝑥)) = ∅)
6160biimpri 216 . . . . . . . . . . 11 ((𝐴 ∖ (𝐹𝑥)) = ∅ → 𝐴 ⊆ (𝐹𝑥))
6259, 61anim12i 587 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → ((𝐹𝑥) ⊆ 𝐴𝐴 ⊆ (𝐹𝑥)))
63 eqss 3582 . . . . . . . . . 10 ((𝐹𝑥) = 𝐴 ↔ ((𝐹𝑥) ⊆ 𝐴𝐴 ⊆ (𝐹𝑥)))
6462, 63sylibr 222 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → (𝐹𝑥) = 𝐴)
65 onss 6860 . . . . . . . . . . . . 13 (𝑥 ∈ On → 𝑥 ⊆ On)
6632, 31nfan 1815 . . . . . . . . . . . . . . . . 17 𝑦(∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑)
67 nfv 1829 . . . . . . . . . . . . . . . . 17 𝑦 𝑥 ⊆ On
6866, 67nfan 1815 . . . . . . . . . . . . . . . 16 𝑦((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On)
69 nfv 1829 . . . . . . . . . . . . . . . . . 18 𝑧(((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦𝑥)
70 ssel 3561 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ⊆ On → (𝑦𝑥𝑦 ∈ On))
71 onss 6860 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → 𝑦 ⊆ On)
72 fndm 5890 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn On → dom 𝐹 = On)
737, 72ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 dom 𝐹 = On
7471, 73syl6sseqr 3614 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ On → 𝑦 ⊆ dom 𝐹)
75 funfvima2 6375 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝐹𝑦 ⊆ dom 𝐹) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
7628, 74, 75sylancr 693 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ On → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
7770, 76syl6 34 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ⊆ On → (𝑦𝑥 → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦))))
7835com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦𝑥 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹𝑦)) ≠ ∅))
7978a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ⊆ On → (𝑦𝑥 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐴 ∖ (𝐹𝑦)) ≠ ∅)))
8070, 79, 44syl10 76 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ⊆ On → (𝑦𝑥 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝜑 → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦))))))
8180imp4a 611 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ⊆ On → (𝑦𝑥 → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
82 eldifn 3694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → ¬ (𝐹𝑦) ∈ (𝐹𝑦))
83 eleq1a 2682 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑧) ∈ (𝐹𝑦) → ((𝐹𝑦) = (𝐹𝑧) → (𝐹𝑦) ∈ (𝐹𝑦)))
8483con3d 146 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑧) ∈ (𝐹𝑦) → (¬ (𝐹𝑦) ∈ (𝐹𝑦) → ¬ (𝐹𝑦) = (𝐹𝑧)))
8582, 84syl5com 31 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → ((𝐹𝑧) ∈ (𝐹𝑦) → ¬ (𝐹𝑦) = (𝐹𝑧)))
8681, 85syl8 73 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ⊆ On → (𝑦𝑥 → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → ((𝐹𝑧) ∈ (𝐹𝑦) → ¬ (𝐹𝑦) = (𝐹𝑧)))))
8786com34 88 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ⊆ On → (𝑦𝑥 → ((𝐹𝑧) ∈ (𝐹𝑦) → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹𝑦) = (𝐹𝑧)))))
8877, 87syldd 69 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ⊆ On → (𝑦𝑥 → (𝑧𝑦 → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹𝑦) = (𝐹𝑧)))))
8988com4r 91 . . . . . . . . . . . . . . . . . . 19 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑦𝑥 → (𝑧𝑦 → ¬ (𝐹𝑦) = (𝐹𝑧)))))
9089imp31 446 . . . . . . . . . . . . . . . . . 18 ((((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦𝑥) → (𝑧𝑦 → ¬ (𝐹𝑦) = (𝐹𝑧)))
9169, 90ralrimi 2939 . . . . . . . . . . . . . . . . 17 ((((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦𝑥) → ∀𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧))
9291ex 448 . . . . . . . . . . . . . . . 16 (((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → (𝑦𝑥 → ∀𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧)))
9368, 92ralrimi 2939 . . . . . . . . . . . . . . 15 (((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧))
9493ex 448 . . . . . . . . . . . . . 14 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧)))
9594ancld 573 . . . . . . . . . . . . 13 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑥 ⊆ On ∧ ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧))))
967tz7.48lem 7401 . . . . . . . . . . . . 13 ((𝑥 ⊆ On ∧ ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧)) → Fun (𝐹𝑥))
9765, 95, 96syl56 35 . . . . . . . . . . . 12 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ∈ On → Fun (𝐹𝑥)))
9897ancoms 467 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑥 ∈ On → Fun (𝐹𝑥)))
9998imp 443 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → Fun (𝐹𝑥))
10099adantr 479 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → Fun (𝐹𝑥))
10126, 64, 1003jca 1234 . . . . . . . 8 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
102101exp41 635 . . . . . . 7 (𝜑 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝑥 ∈ On → ((𝐴 ∖ (𝐹𝑥)) = ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))))
103102com23 83 . . . . . 6 (𝜑 → (𝑥 ∈ On → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → ((𝐴 ∖ (𝐹𝑥)) = ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))))
104103com34 88 . . . . 5 (𝜑 → (𝑥 ∈ On → ((𝐴 ∖ (𝐹𝑥)) = ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))))
105104imp4a 611 . . . 4 (𝜑 → (𝑥 ∈ On → (((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))))
10625, 105reximdai 2994 . . 3 (𝜑 → (∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))
10723, 106syld 45 . 2 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))
108107impcom 444 1 ((𝐴𝐵𝜑) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  wss 3539  c0 3873  ccnv 5027  dom cdm 5028  cres 5030  cima 5031  Oncon0 5626  Fun wfun 5784   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798
This theorem is referenced by:  tz7.49c  7406
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