Theorem fmpodg 48704

Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Zhi Wang, 29-Sep-2025.)

Hypotheses

Ref	Expression
fmpodg.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))
fmpodg.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑆)
fmpodg.3	⊢ (𝜑 → 𝑅 = (𝐴 × 𝐵))

Assertion

Ref	Expression
fmpodg	⊢ (𝜑 → 𝐹:𝑅⟶𝑆)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fmpodg

Step	Hyp	Ref	Expression
1		fmpodg.2	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑆)
2	1	ralrimivva 3189	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑆)
3		eqid 2734	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fmpo 8074	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶):(𝐴 × 𝐵)⟶𝑆)
5	2, 4	sylib 218	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶):(𝐴 × 𝐵)⟶𝑆)
6		fmpodg.1	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))
7		fmpodg.3	. . 3 ⊢ (𝜑 → 𝑅 = (𝐴 × 𝐵))
8	6, 7	feq12d 6703	. 2 ⊢ (𝜑 → (𝐹:𝑅⟶𝑆 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶):(𝐴 × 𝐵)⟶𝑆))
9	5, 8	mpbird 257	1 ⊢ (𝜑 → 𝐹:𝑅⟶𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050   × cxp 5663  ⟶wf 6536   ∈ cmpo 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-oprab 7416  df-mpo 7417  df-1st 7995  df-2nd 7996

This theorem is referenced by:  fmpod  48705  fucof21  48994