Theorem xpco2 49520

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 6111	. 2 ⊢ Rel ((𝐵 × 𝐶) ∘ 𝐹)
2		relxp 5680	. 2 ⊢ Rel (𝐴 × 𝐶)
3		vex 3467	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 3467	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
5	3, 4	breldm 5899	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6	5	ad2antrl 740	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ dom 𝐹)
7		fdm 6716	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
8	7	adantr 485	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → dom 𝐹 = 𝐴)
9	6, 8	eleqtrd 2871	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑥 ∈ 𝐴)
10		brxp 5711	. . . . . . . . . 10 ⊢ (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
11	10	simprbi 502	. . . . . . . . 9 ⊢ (𝑧(𝐵 × 𝐶)𝑦 → 𝑦 ∈ 𝐶)
12	11	ad2antll 741	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → 𝑦 ∈ 𝐶)
13	9, 12	jca 520	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
14	13	ex 417	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
15	14	exlimdv 1960	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
16	15	imp 411	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
17		ffvelcdm 7077	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
18	17	adantrr 729	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥) ∈ 𝐵)
19		ffvbr 49519	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥))
20	19	adantrr 729	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑥𝐹(𝐹‘𝑥))
21		simprr 784	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
22		brxp 5711	. . . . . . 7 ⊢ ((𝐹‘𝑥)(𝐵 × 𝐶)𝑦 ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
23	18, 21, 22	sylanbrc 594	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)
24	20, 23	jca 520	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
25		breq2 5117	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑥𝐹𝑧 ↔ 𝑥𝐹(𝐹‘𝑥)))
26		breq1 5116	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧(𝐵 × 𝐶)𝑦 ↔ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦))
27	25, 26	anbi12d 643	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥𝐹(𝐹‘𝑥) ∧ (𝐹‘𝑥)(𝐵 × 𝐶)𝑦)))
28	18, 24, 27	spcedv 3566	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
29	16, 28	impbida 812	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
30		vex 3467	. . . 4 ⊢ 𝑦 ∈ V
31	3, 30	brco 5857	. . 3 ⊢ (𝑥((𝐵 × 𝐶) ∘ 𝐹)𝑦 ↔ ∃𝑧(𝑥𝐹𝑧 ∧ 𝑧(𝐵 × 𝐶)𝑦))
32		brxp 5711	. . 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 5781	1 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   class class class wbr 5113   × cxp 5660  dom cdm 5662   ∘ ccom 5666  ⟶wf 6533  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is referenced by:  prcofdiag  50057