Theorem funexw 7976

Description: Weak version of funex 7239 that holds without ax-rep 5279. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Assertion

Ref	Expression
funexw	⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V)

Proof of Theorem funexw

Step	Hyp	Ref	Expression
1		xpexg 7770	. . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
2	1	3adant1 1131	. 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → (dom 𝐹 × ran 𝐹) ∈ V)
3		funrel 6583	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
4		relssdmrn 6288	. . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5	3, 4	syl 17	. . 3 ⊢ (Fun 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
6	5	3ad2ant1 1134	. 2 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
7	2, 6	ssexd 5324	1 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ 𝐶) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  Vcvv 3480   ⊆ wss 3951   × cxp 5683  dom cdm 5685  ran crn 5686  Rel wrel 5690  Fun wfun 6555

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-fun 6563

This theorem is referenced by:  mptexw  7977  mpoexw  8103  seqexw  14058