Theorem fmpodg 48786

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 3182	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑆)
3		eqid 2730	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fmpo 8056	. . 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 6683	. 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 1540   ∈ wcel 2109  ∀wral 3046   × cxp 5644  ⟶wf 6515   ∈ cmpo 7396

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fv 6527  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978

This theorem is referenced by:  fmpod  48787  fucof21  49242