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Theorem tz7.48-1 7690
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 3352 . . . . 5 𝑦 ∈ V
21elrn2 5503 . . . 4 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
3 vex 3352 . . . . . . . . 9 𝑥 ∈ V
43, 1opeldm 5466 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐹)
5 tz7.48.1 . . . . . . . . 9 𝐹 Fn On
6 fndm 6130 . . . . . . . . 9 (𝐹 Fn On → dom 𝐹 = On)
75, 6ax-mp 5 . . . . . . . 8 dom 𝐹 = On
84, 7syl6eleq 2859 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ On)
98ancri 531 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10 fnopfvb 6378 . . . . . . . 8 ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
115, 10mpan 662 . . . . . . 7 (𝑥 ∈ On → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
1211pm5.32i 556 . . . . . 6 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
139, 12sylibr 224 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
1413eximi 1909 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
152, 14sylbi 207 . . 3 (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
16 nfra1 3089 . . . 4 𝑥𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))
17 nfv 1994 . . . 4 𝑥 𝑦𝐴
18 rsp 3077 . . . . 5 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
19 eldifi 3881 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐴)
20 eleq1 2837 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2119, 20syl5ibcom 235 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2221imim2i 16 . . . . . 6 ((𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
2322impd 396 . . . . 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 2242 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2615, 25syl5 34 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
2726ssrdv 3756 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wex 1851  wcel 2144  wral 3060  cdif 3718  wss 3721  cop 4320  dom cdm 5249  ran crn 5250  cima 5252  Oncon0 5866   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  tz7.48-3  7691
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