Theorem xpco2 49112

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 6067	. 2 ⊢ Rel ((𝐵 × 𝐶) ∘ 𝐹)
2		relxp 5642	. 2 ⊢ Rel (𝐴 × 𝐶)
3		vex 3444	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 3444	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
5	3, 4	breldm 5857	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6	5	ad2antrl 728	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ dom 𝐹)
7		fdm 6671	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → dom 𝐹 = 𝐴)
9	6, 8	eleqtrd 2838	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ 𝐴)
10		brxp 5673	. . . . . . . . . 10 ⊢ (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
11	10	simprbi 496	. . . . . . . . 9 ⊢ (𝑧(𝐵 × 𝐶)𝑦 → 𝑦 ∈ 𝐶)
12	11	ad2antll 729	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑦 ∈ 𝐶)
13	9, 12	jca 511	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
14	13	ex 412	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
15	14	exlimdv 1934	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
16	15	imp 406	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
17		ffvelcdm 7026	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
18	17	adantrr 717	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥) ∈ 𝐵)
19		ffvbr 49111	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
20	19	adantrr 717	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑥𝐹(𝐹‘𝑥))
21		simprr 772	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
22		brxp 5673	. . . . . . 7 ⊢ ((𝐹‘𝑥)(𝐵 × 𝐶)𝑦 ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
23	18, 21, 22	sylanbrc 583	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)
24	20, 23	jca 511	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
25		breq2 5102	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(𝐹‘𝑥)))
26		breq1 5101	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
27	25, 26	anbi12d 632	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)))
28	18, 24, 27	spcedv 3552	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
29	16, 28	impbida 800	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
30		vex 3444	. . . 4 ⊢ 𝑦 ∈ V
31	3, 30	brco 5819	. . 3 ⊢ (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
32		brxp 5673	. . 3 ⊢ (𝑥(𝐴 × 𝐶)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
33	29, 31, 32	3bitr4g 314	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ 𝑥(𝐴 × 𝐶)𝑦))
34	1, 2, 33	eqbrrdiv 5743	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   class class class wbr 5098   × cxp 5622  dom cdm 5624   ∘ ccom 5628  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  prcofdiag  49649