Theorem fmpodg 48715

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 3202	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑆)
3		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fmpo 8101	. . 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 6732	. 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 2108  ∀wral 3061   × cxp 5691  ⟶wf 6565   ∈ cmpo 7440

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fv 6577  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023

This theorem is referenced by:  fmpod  48716  fucof21  48914