Theorem tz7.48-1 8481

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 (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz7.48-1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elrn2 5881	. . . 4 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹)
3		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5896	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
5		tz7.48.1	. . . . . . . . 9 ⊢ 𝐹 Fn On
6	5	fndmi 6665	. . . . . . . 8 ⊢ dom 𝐹 = On
7	4, 6	eleqtrdi 2874	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ On)
8	7	ancri 559	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9		fnopfvb 6959	. . . . . . . 8 ⊢ ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10	5, 9	mpan 703	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	10	pm5.32i 585	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
12	8, 11	sylibr 237	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
13	12	eximi 1868	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
14	2, 13	sylbi 220	. . 3 ⊢ (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
15		nfra1 3290	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))
16		nfv 1949	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
17		rsp 3254	. . . . 5 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
18		eldifi 4070	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐹‘𝑥) ∈ 𝐴)
19		eleq1 2852	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
20	18, 19	syl5ibcom 248	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
21	20	imim2i 17	. . . . . 6 ⊢ ((𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴)))
22	21	impd 416	. . . . 5 ⊢ ((𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
23	17, 22	syl 18	. . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
24	15, 16, 23	exlimd 2258	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
25	14, 24	syl5 35	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐴))
26	25	ssrdv 3919	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∖ cdif 4048  ⟨cop 4838  dom cdm 5655  ran crn 5656   “ cima 5658  Oncon0 6380   Fn wfn 6553  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-fv 6566

This theorem is used by:  tz7.48-3  8482