Theorem xpco2 50110

Description: Composition of a Cartesian product with a function. (Contributed by Zhi Wang, 25-Nov-2025.)

Assertion

Ref	Expression
xpco2	⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))

Proof of Theorem xpco2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6110	. 2 ⊢ Rel ((𝐵 × 𝐶) ∘ 𝐹)
2		relxp 5673	. 2 ⊢ Rel (𝐴 × 𝐶)
3		vex 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 3458	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
5	3, 4	breldm 5895	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6	5	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ dom 𝐹)
7		fdm 6730	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
8	7	adantr 486	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → dom 𝐹 = 𝐴)
9	6, 8	eleqtrd 2866	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ 𝐴)
10		brxp 5704	. . . . . . . . . 10 ⊢ (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
11	10	simprbi 503	. . . . . . . . 9 ⊢ (𝑧(𝐵 × 𝐶)𝑦 → 𝑦 ∈ 𝐶)
12	11	ad2antll 742	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑦 ∈ 𝐶)
13	9, 12	jca 521	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
14	13	ex 418	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
15	14	exlimdv 1967	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
16	15	imp 412	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
17		ffvelcdm 7092	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
18	17	adantrr 730	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥) ∈ 𝐵)
19		ffvbr 50109	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
20	19	adantrr 730	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑥𝐹(𝐹‘𝑥))
21		simprr 785	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
22		brxp 5704	. . . . . . 7 ⊢ ((𝐹‘𝑥)(𝐵 × 𝐶)𝑦 ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
23	18, 21, 22	sylanbrc 595	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)
24	20, 23	jca 521	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
25		breq2 5110	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(𝐹‘𝑥)))
26		breq1 5109	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
27	25, 26	anbi12d 644	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)))
28	18, 24, 27	spcedv 3556	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
29	16, 28	impbida 813	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
30		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
31	3, 30	brco 5853	. . 3 ⊢ (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
32		brxp 5704	. . 3 ⊢ (𝑥(𝐴 × 𝐶)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
33	29, 31, 32	3bitr4g 317	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ 𝑥(𝐴 × 𝐶)𝑦))
34	1, 2, 33	eqbrrdiv 5775	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146   class class class wbr 5106   × cxp 5653  dom cdm 5655   ∘ ccom 5659  ⟶wf 6543  ‘cfv 6547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555

This theorem is used by:  prcofdiag  50671