Theorem fabexg 7976

Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.) (Proof shortened by AV, 9-Jun-2025.)

Hypothesis

Ref	Expression
fabexg.1	⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}

Assertion

Ref	Expression
fabexg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)

Proof of Theorem fabexg

Step	Hyp	Ref	Expression
1		elex 3509	. 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		elex 3509	. 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
3		fabexg.1	. . 3 ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}
4		simprl 770	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝑥:𝐴⟶𝐵 ∧ 𝜑)) → 𝑥:𝐴⟶𝐵)
5		simpl 482	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
6		simpr 484	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
7	4, 5, 6	fabexd 7975	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} ∈ V)
8	3, 7	eqeltrid 2848	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
9	1, 2, 8	syl2an 595	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  fabex  7978  mapex  7979  f1oabexgOLD  7981  elghomlem1OLD  37845