Theorem tz7.48-1 8418

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 3461	. . . . 5 ⊢ 𝑦 ∈ V
2	1	elrn2 5873	. . . 4 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐹)
3		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
4	3, 1	opeldm 5888	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
5		tz7.48.1	. . . . . . . . 9 ⊢ 𝐹 Fn On
6	5	fndmi 6629	. . . . . . . 8 ⊢ dom 𝐹 = On
7	4, 6	eleqtrdi 2875	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ On)
8	7	ancri 558	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
9		fnopfvb 6922	. . . . . . . 8 ⊢ ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10	5, 9	mpan 702	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
11	10	pm5.32i 584	. . . . . 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 1858	. . . 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 3289	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))
16		nfv 1937	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
17		rsp 3253	. . . . 5 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
18		eldifi 4087	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐹‘𝑥) ∈ 𝐴)
19		eleq1 2853	. . . . . . . 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 415	. . . . 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 2256	. . 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 3945	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079   ∖ cdif 3904   ⊆ wss 3907  ⟨cop 4591  dom cdm 5652  ran crn 5653   “ cima 5655  Oncon0 6350   Fn wfn 6520  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533

This theorem is referenced by:  tz7.48-3  8419