Theorem tz7.48-1 7538

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 3203	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elrn2 5365	. . . 4 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹)
3		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5328	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
5		tz7.48.1	. . . . . . . . 9 ⊢ 𝐹 Fn On
6		fndm 5990	. . . . . . . . 9 ⊢ (𝐹 Fn On → dom 𝐹 = On)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ dom 𝐹 = On
8	4, 7	syl6eleq 2711	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ On)
9	8	ancri 575	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10		fnopfvb 6237	. . . . . . . 8 ⊢ ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	5, 10	mpan 706	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
12	11	pm5.32i 669	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
13	9, 12	sylibr 224	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
14	13	eximi 1762	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
15	2, 14	sylbi 207	. . 3 ⊢ (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
16		nfra1 2941	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))
17		nfv 1843	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
18		rsp 2929	. . . . 5 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
19		eldifi 3732	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐹‘𝑥) ∈ 𝐴)
20		eleq1 2689	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
21	19, 20	syl5ibcom 235	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
22	21	imim2i 16	. . . . . 6 ⊢ ((𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴)))
23	22	impd 447	. . . . 5 ⊢ ((𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
24	18, 23	syl 17	. . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
25	16, 17, 24	exlimd 2087	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐴))
26	15, 25	syl5 34	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐴))
27	26	ssrdv 3609	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912   ∖ cdif 3571   ⊆ wss 3574  ⟨cop 4183  dom cdm 5114  ran crn 5115   “ cima 5117  Oncon0 5723   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  tz7.48-3  7539