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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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