Theorem resfunexgALT 7890

Description: Alternate proof of resfunexg 7159, shorter but requiring ax-pow 5294 and ax-un 7678. (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 5966	. . . 4 ⊢ (𝐵 ∈ 𝐶 → dom (𝐴 ↾ 𝐵) ∈ V)
2	1	adantl 482	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → dom (𝐴 ↾ 𝐵) ∈ V)
3		df-ima 5631	. . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
4		funimaexg 6572	. . . 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 516	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V))
7		xpexg 7693	. 2 ⊢ ((dom (𝐴 ↾ 𝐵) ∈ V ∧ ran (𝐴 ↾ 𝐵) ∈ V) → (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V)
8		relres 5957	. . . 4 ⊢ Rel (𝐴 ↾ 𝐵)
9		relssdmrn 6220	. . . 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 5251	. . 3 ⊢ (((𝐴 ↾ 𝐵) ⊆ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∧ (dom (𝐴 ↾ 𝐵) × ran (𝐴 ↾ 𝐵)) ∈ V) → (𝐴 ↾ 𝐵) ∈ V)
12	10, 11	mpan 696	. 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 2119  Vcvv 3431   ⊆ wss 3883   × cxp 5616  dom cdm 5618  ran crn 5619   ↾ cres 5620   “ cima 5621  Rel wrel 5623  Fun wfun 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-fun 6487

This theorem is referenced by: (None)