Theorem resfunexgALT 7790

Description: Alternate proof of resfunexg 7091, shorter but requiring ax-pow 5288 and ax-un 7588. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
resfunexgALT	⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Proof of Theorem resfunexgALT

Step	Hyp	Ref	Expression
1		dmresexg 5915	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
2	1	adantl 482	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
3		df-ima 5602	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
4		funimaexg 6520	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
5	3, 4	eqeltrrid 2844	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ↾ 𝐵) ∈ V)
6	2, 5	jca 512	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V))
7		xpexg 7600	. 2 ⊢ ((dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V) → (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V)
8		relres 5920	. . . 4 ⊢ Rel (𝐴 ↾ 𝐵)
9		relssdmrn 6172	. . . 4 ⊢ (Rel (𝐴 ↾ 𝐵) → (𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)))
10	8, 9	ax-mp 5	. . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵))
11		ssexg 5247	. . 3 ⊢ (((𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∧ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V) → (𝐴 ↾ 𝐵) ∈ V)
12	10, 11	mpan 687	. 2 ⊢ ((dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V → (𝐴 ↾ 𝐵) ∈ V)
13	6, 7, 12	3syl 18	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887   × cxp 5587  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592  Rel wrel 5594  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435

This theorem is referenced by: (None)