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Theorem tz7.48-1 7402
Description: Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
Hypothesis
Ref Expression
tz7.48.1 𝐹 Fn On
Assertion
Ref Expression
tz7.48-1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz7.48-1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3175 . . . . 5 𝑦 ∈ V
21elrn2 5273 . . . 4 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
3 vex 3175 . . . . . . . . 9 𝑥 ∈ V
43, 1opeldm 5237 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐹)
5 tz7.48.1 . . . . . . . . 9 𝐹 Fn On
6 fndm 5890 . . . . . . . . 9 (𝐹 Fn On → dom 𝐹 = On)
75, 6ax-mp 5 . . . . . . . 8 dom 𝐹 = On
84, 7syl6eleq 2697 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ On)
98ancri 572 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10 fnopfvb 6132 . . . . . . . 8 ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
115, 10mpan 701 . . . . . . 7 (𝑥 ∈ On → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
1211pm5.32i 666 . . . . . 6 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
139, 12sylibr 222 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
1413eximi 1751 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
152, 14sylbi 205 . . 3 (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
16 nfra1 2924 . . . 4 𝑥𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))
17 nfv 1829 . . . 4 𝑥 𝑦𝐴
18 rsp 2912 . . . . 5 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
19 eldifi 3693 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐴)
20 eleq1 2675 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2119, 20syl5ibcom 233 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2221imim2i 16 . . . . . 6 ((𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
2322impd 445 . . . . 5 ((𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2418, 23syl 17 . . . 4 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2516, 17, 24exlimd 2073 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2615, 25syl5 33 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
2726ssrdv 3573 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wral 2895  cdif 3536  wss 3539  cop 4130  dom cdm 5028  ran crn 5029  cima 5031  Oncon0 5626   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-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  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-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  tz7.48-3  7403
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