Theorem fmpodg 49344

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 3180	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑆)
3		eqid 2736	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fmpo 8021	. . 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 6656	. 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 1542   ∈ wcel 2114  ∀wral 3051   × cxp 5629  ⟶wf 6494   ∈ cmpo 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943

This theorem is referenced by:  fmpod  49345  fucof21  49822