Theorem tz7.48-1 8439

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 3478	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elrn2 5890	. . . 4 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹)
3		vex 3478	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5905	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
5		tz7.48.1	. . . . . . . . 9 ⊢ 𝐹 Fn On
6	5	fndmi 6650	. . . . . . . 8 ⊢ dom 𝐹 = On
7	4, 6	eleqtrdi 2843	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ On)
8	7	ancri 550	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9		fnopfvb 6942	. . . . . . . 8 ⊢ ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10	5, 9	mpan 688	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	10	pm5.32i 575	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
12	8, 11	sylibr 233	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
13	12	eximi 1837	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
14	2, 13	sylbi 216	. . 3 ⊢ (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
15		nfra1 3281	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))
16		nfv 1917	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
17		rsp 3244	. . . . 5 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
18		eldifi 4125	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐹‘𝑥) ∈ 𝐴)
19		eleq1 2821	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
20	18, 19	syl5ibcom 244	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
21	20	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴)))
22	21	impd 411	. . . . 5 ⊢ ((𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
23	17, 22	syl 17	. . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
24	15, 16, 23	exlimd 2211	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
25	14, 24	syl5 34	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐴))
26	25	ssrdv 3987	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061   ∖ cdif 3944   ⊆ wss 3947  ⟨cop 4633  dom cdm 5675  ran crn 5676   “ cima 5678  Oncon0 6361   Fn wfn 6535  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-fv 6548

This theorem is referenced by:  tz7.48-3  8440