Theorem resfunexgALT 7935

Description: Alternate proof of resfunexg 7196, shorter but requiring ax-pow 5328 and ax-un 7718. (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 5993	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
2	1	adantl 481	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
3		df-ima 5659	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
4		funimaexg 6611	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
5	3, 4	eqeltrrid 2834	. . 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 7733	. 2 ⊢ ((dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V) → (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V)
8		relres 5984	. . . 4 ⊢ Rel (𝐴 ↾ 𝐵)
9		relssdmrn 6249	. . . 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 5286	. . 3 ⊢ (((𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∧ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V) → (𝐴 ↾ 𝐵) ∈ V)
12	10, 11	mpan 690	. 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 2109  Vcvv 3455   ⊆ wss 3922   × cxp 5644  dom cdm 5646  ran crn 5647   ↾ cres 5648   “ cima 5649  Rel wrel 5651  Fun wfun 6513

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-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-fun 6521

This theorem is referenced by: (None)