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Theorem tz7.48-1 8390
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 3450 . . . . 5 𝑦 ∈ V
21elrn2 5849 . . . 4 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
3 vex 3450 . . . . . . . . 9 𝑥 ∈ V
43, 1opeldm 5864 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐹)
5 tz7.48.1 . . . . . . . . 9 𝐹 Fn On
65fndmi 6607 . . . . . . . 8 dom 𝐹 = On
74, 6eleqtrdi 2848 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ On)
87ancri 551 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9 fnopfvb 6897 . . . . . . . 8 ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
105, 9mpan 689 . . . . . . 7 (𝑥 ∈ On → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
1110pm5.32i 576 . . . . . 6 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
128, 11sylibr 233 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
1312eximi 1838 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
142, 13sylbi 216 . . 3 (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
15 nfra1 3268 . . . 4 𝑥𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))
16 nfv 1918 . . . 4 𝑥 𝑦𝐴
17 rsp 3231 . . . . 5 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
18 eldifi 4087 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐴)
19 eleq1 2826 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2018, 19syl5ibcom 244 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2120imim2i 16 . . . . . 6 ((𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
2221impd 412 . . . . 5 ((𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2317, 22syl 17 . . . 4 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2415, 16, 23exlimd 2212 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2514, 24syl5 34 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
2625ssrdv 3951 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3065  cdif 3908  wss 3911  cop 4593  dom cdm 5634  ran crn 5635  cima 5637  Oncon0 6318   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  tz7.48-3  8391
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