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Theorem tz7.49 7748
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 2938 . . . . . . . . 9 ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹𝑥)) = ∅)
21ralbii 3127 . . . . . . . 8 (∀𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) ≠ ∅ ↔ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅)
3 tz7.49.2 . . . . . . . . 9 (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
4 ralim 3095 . . . . . . . . 9 (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (∀𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
53, 4sylbi 208 . . . . . . . 8 (𝜑 → (∀𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) ≠ ∅ → ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
62, 5syl5bir 234 . . . . . . 7 (𝜑 → (∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅ → ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
7 tz7.49.1 . . . . . . . . 9 𝐹 Fn On
87tz7.48-3 7747 . . . . . . . 8 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ 𝐴 ∈ V)
9 elex 3365 . . . . . . . 8 (𝐴𝐵𝐴 ∈ V)
108, 9nsyl3 135 . . . . . . 7 (𝐴𝐵 → ¬ ∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
116, 10nsyli 156 . . . . . 6 (𝜑 → (𝐴𝐵 → ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅))
12 dfrex2 3142 . . . . . 6 (∃𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) = ∅ ↔ ¬ ∀𝑥 ∈ On ¬ (𝐴 ∖ (𝐹𝑥)) = ∅)
1311, 12syl6ibr 243 . . . . 5 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) = ∅))
14 imaeq2 5646 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1514difeq2d 3892 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 ∖ (𝐹𝑥)) = (𝐴 ∖ (𝐹𝑦)))
1615eqeq1d 2767 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹𝑥)) = ∅ ↔ (𝐴 ∖ (𝐹𝑦)) = ∅))
1716onminex 7209 . . . . 5 (∃𝑥 ∈ On (𝐴 ∖ (𝐹𝑥)) = ∅ → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅))
1813, 17syl6 35 . . . 4 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅)))
19 df-ne 2938 . . . . . . 7 ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ ↔ ¬ (𝐴 ∖ (𝐹𝑦)) = ∅)
2019ralbii 3127 . . . . . 6 (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ↔ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅)
2120anbi2i 616 . . . . 5 (((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ↔ ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅))
2221rexbii 3188 . . . 4 (∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ↔ ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 ¬ (𝐴 ∖ (𝐹𝑦)) = ∅))
2318, 22syl6ibr 243 . . 3 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅)))
24 nfra1 3088 . . . . 5 𝑥𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))
253, 24nfxfr 1948 . . . 4 𝑥𝜑
26 simpllr 793 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅)
27 fnfun 6168 . . . . . . . . . . . . . . . . 17 (𝐹 Fn On → Fun 𝐹)
287, 27ax-mp 5 . . . . . . . . . . . . . . . 16 Fun 𝐹
29 fvelima 6441 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ∈ (𝐹𝑥)) → ∃𝑦𝑥 (𝐹𝑦) = 𝑧)
3028, 29mpan 681 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝐹𝑥) → ∃𝑦𝑥 (𝐹𝑦) = 𝑧)
31 nfv 2009 . . . . . . . . . . . . . . . . 17 𝑦𝜑
32 nfra1 3088 . . . . . . . . . . . . . . . . 17 𝑦𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅
3331, 32nfan 1998 . . . . . . . . . . . . . . . 16 𝑦(𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅)
34 nfv 2009 . . . . . . . . . . . . . . . 16 𝑦(𝑥 ∈ On → 𝑧𝐴)
35 rsp 3076 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝑦𝑥 → (𝐴 ∖ (𝐹𝑦)) ≠ ∅))
3635adantld 484 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝐴 ∖ (𝐹𝑦)) ≠ ∅))
37 onelon 5935 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3815neeq1d 2996 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑦 → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑦)) ≠ ∅))
39 fveq2 6379 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
4039, 15eleq12d 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) ↔ (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦))))
4138, 40imbi12d 335 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦 → (((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) ↔ ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4241rspcv 3458 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ On → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
433, 42syl5bi 233 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (𝜑 → ((𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4443com23 86 . . . . . . . . . . . . . . . . . . . . . . 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 87 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
4847imp 395 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦))))
4948expcomd 406 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑦𝑥 → (𝑥 ∈ On → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
50 eldifi 3896 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → (𝐹𝑦) ∈ 𝐴)
51 eleq1 2832 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ 𝐴𝑧𝐴))
5250, 51syl5ibcom 236 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → ((𝐹𝑦) = 𝑧𝑧𝐴))
5349, 52syl8 76 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑦𝑥 → (𝑥 ∈ On → ((𝐹𝑦) = 𝑧𝑧𝐴))))
5453com34 91 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑦𝑥 → ((𝐹𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧𝐴))))
5533, 34, 54rexlimd 3173 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (∃𝑦𝑥 (𝐹𝑦) = 𝑧 → (𝑥 ∈ On → 𝑧𝐴)))
5630, 55syl5 34 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑧 ∈ (𝐹𝑥) → (𝑥 ∈ On → 𝑧𝐴)))
5756com23 86 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑥 ∈ On → (𝑧 ∈ (𝐹𝑥) → 𝑧𝐴)))
5857imp 395 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝑧 ∈ (𝐹𝑥) → 𝑧𝐴))
5958ssrdv 3769 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → (𝐹𝑥) ⊆ 𝐴)
60 ssdif0 4108 . . . . . . . . . . . 12 (𝐴 ⊆ (𝐹𝑥) ↔ (𝐴 ∖ (𝐹𝑥)) = ∅)
6160biimpri 219 . . . . . . . . . . 11 ((𝐴 ∖ (𝐹𝑥)) = ∅ → 𝐴 ⊆ (𝐹𝑥))
6259, 61anim12i 606 . . . . . . . . . 10 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → ((𝐹𝑥) ⊆ 𝐴𝐴 ⊆ (𝐹𝑥)))
63 eqss 3778 . . . . . . . . . 10 ((𝐹𝑥) = 𝐴 ↔ ((𝐹𝑥) ⊆ 𝐴𝐴 ⊆ (𝐹𝑥)))
6462, 63sylibr 225 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → (𝐹𝑥) = 𝐴)
65 onss 7192 . . . . . . . . . . . . 13 (𝑥 ∈ On → 𝑥 ⊆ On)
6632, 31nfan 1998 . . . . . . . . . . . . . . . . 17 𝑦(∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑)
67 nfv 2009 . . . . . . . . . . . . . . . . 17 𝑦 𝑥 ⊆ On
6866, 67nfan 1998 . . . . . . . . . . . . . . . 16 𝑦((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On)
69 nfv 2009 . . . . . . . . . . . . . . . . . 18 𝑧(((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦𝑥)
70 ssel 3757 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ⊆ On → (𝑦𝑥𝑦 ∈ On))
71 onss 7192 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → 𝑦 ⊆ On)
72 fndm 6170 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn On → dom 𝐹 = On)
737, 72ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 dom 𝐹 = On
7471, 73syl6sseqr 3814 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ On → 𝑦 ⊆ dom 𝐹)
75 funfvima2 6690 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝐹𝑦 ⊆ dom 𝐹) → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
7628, 74, 75sylancr 581 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ On → (𝑧𝑦 → (𝐹𝑧) ∈ (𝐹𝑦)))
7770, 76syl6 35 . . . . . . . . . . . . . . . . . . . . 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 79 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ⊆ On → (𝑦𝑥 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝜑 → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦))))))
8180imp4a 413 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ⊆ On → (𝑦𝑥 → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)))))
82 eldifn 3897 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → ¬ (𝐹𝑦) ∈ (𝐹𝑦))
83 eleq1a 2839 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑧) ∈ (𝐹𝑦) → ((𝐹𝑦) = (𝐹𝑧) → (𝐹𝑦) ∈ (𝐹𝑦)))
8483con3d 149 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑧) ∈ (𝐹𝑦) → (¬ (𝐹𝑦) ∈ (𝐹𝑦) → ¬ (𝐹𝑦) = (𝐹𝑧)))
8582, 84syl5com 31 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑦) ∈ (𝐴 ∖ (𝐹𝑦)) → ((𝐹𝑧) ∈ (𝐹𝑦) → ¬ (𝐹𝑦) = (𝐹𝑧)))
8681, 85syl8 76 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ⊆ On → (𝑦𝑥 → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → ((𝐹𝑧) ∈ (𝐹𝑦) → ¬ (𝐹𝑦) = (𝐹𝑧)))))
8786com34 91 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ⊆ On → (𝑦𝑥 → ((𝐹𝑧) ∈ (𝐹𝑦) → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹𝑦) = (𝐹𝑧)))))
8877, 87syldd 72 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ⊆ On → (𝑦𝑥 → (𝑧𝑦 → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → ¬ (𝐹𝑦) = (𝐹𝑧)))))
8988com4r 94 . . . . . . . . . . . . . . . . . . 19 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑦𝑥 → (𝑧𝑦 → ¬ (𝐹𝑦) = (𝐹𝑧)))))
9089imp31 408 . . . . . . . . . . . . . . . . . 18 ((((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦𝑥) → (𝑧𝑦 → ¬ (𝐹𝑦) = (𝐹𝑧)))
9169, 90ralrimi 3104 . . . . . . . . . . . . . . . . 17 ((((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) ∧ 𝑦𝑥) → ∀𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧))
9291ex 401 . . . . . . . . . . . . . . . 16 (((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → (𝑦𝑥 → ∀𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧)))
9368, 92ralrimi 3104 . . . . . . . . . . . . . . 15 (((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) ∧ 𝑥 ⊆ On) → ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧))
9493ex 401 . . . . . . . . . . . . . 14 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧)))
9594ancld 546 . . . . . . . . . . . . 13 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ⊆ On → (𝑥 ⊆ On ∧ ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧))))
967tz7.48lem 7744 . . . . . . . . . . . . 13 ((𝑥 ⊆ On ∧ ∀𝑦𝑥𝑧𝑦 ¬ (𝐹𝑦) = (𝐹𝑧)) → Fun (𝐹𝑥))
9765, 95, 96syl56 36 . . . . . . . . . . . 12 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ 𝜑) → (𝑥 ∈ On → Fun (𝐹𝑥)))
9897ancoms 450 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (𝑥 ∈ On → Fun (𝐹𝑥)))
9998imp 395 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) → Fun (𝐹𝑥))
10099adantr 472 . . . . . . . . 9 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → Fun (𝐹𝑥))
10126, 64, 1003jca 1158 . . . . . . . 8 ((((𝜑 ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹𝑥)) = ∅) → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
102101exp41 425 . . . . . . 7 (𝜑 → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (𝑥 ∈ On → ((𝐴 ∖ (𝐹𝑥)) = ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))))
103102com23 86 . . . . . 6 (𝜑 → (𝑥 ∈ On → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → ((𝐴 ∖ (𝐹𝑥)) = ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))))
104103com34 91 . . . . 5 (𝜑 → (𝑥 ∈ On → ((𝐴 ∖ (𝐹𝑥)) = ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))))
105104imp4a 413 . . . 4 (𝜑 → (𝑥 ∈ On → (((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))))
10625, 105reximdai 3158 . . 3 (𝜑 → (∃𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) = ∅ ∧ ∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))
10723, 106syld 47 . 2 (𝜑 → (𝐴𝐵 → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥))))
108107impcom 396 1 ((𝐴𝐵𝜑) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3731  wss 3734  c0 4081  ccnv 5278  dom cdm 5279  cres 5281  cima 5282  Oncon0 5910  Fun wfun 6064   Fn wfn 6065  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078
This theorem is referenced by:  tz7.49c  7749
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