Theorem tz7.48-1 8388

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 3448	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elrn2 5846	. . . 4 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹)
3		vex 3448	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5861	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
5		tz7.48.1	. . . . . . . . 9 ⊢ 𝐹 Fn On
6	5	fndmi 6604	. . . . . . . 8 ⊢ dom 𝐹 = On
7	4, 6	eleqtrdi 2838	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ On)
8	7	ancri 549	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9		fnopfvb 6894	. . . . . . . 8 ⊢ ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10	5, 9	mpan 690	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	10	pm5.32i 574	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
12	8, 11	sylibr 234	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
13	12	eximi 1835	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
14	2, 13	sylbi 217	. . 3 ⊢ (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹‘𝑥) = 𝑦))
15		nfra1 3259	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))
16		nfv 1914	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
17		rsp 3223	. . . . 5 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
18		eldifi 4090	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐹‘𝑥) ∈ 𝐴)
19		eleq1 2816	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
20	18, 19	syl5ibcom 245	. . . . . . 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 410	. . . . 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 2219	. . 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 3949	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3908   ⊆ wss 3911  ⟨cop 4591  dom cdm 5631  ran crn 5632   “ cima 5634  Oncon0 6320   Fn wfn 6494  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507

This theorem is referenced by:  tz7.48-3  8389