Theorem resfunexgALT 7894

Description: Alternate proof of resfunexg 7163, shorter but requiring ax-pow 5302 and ax-un 7682. (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 5973	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
2	1	adantl 481	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
3		df-ima 5637	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
4		funimaexg 6579	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
5	3, 4	eqeltrrid 2842	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → ran (𝐴 ↾ 𝐵) ∈ V)
6	2, 5	jca 511	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V))
7		xpexg 7697	. 2 ⊢ ((dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V) → (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V)
8		relres 5964	. . . 4 ⊢ Rel (𝐴 ↾ 𝐵)
9		relssdmrn 6227	. . . 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 5260	. . 3 ⊢ (((𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∧ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V) → (𝐴 ↾ 𝐵) ∈ V)
12	10, 11	mpan 691	. 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 395   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890   × cxp 5622  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627  Rel wrel 5629  Fun wfun 6486

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494

This theorem is referenced by: (None)