Theorem fmpocos 42252

Description: Composition of two functions. Variation of fmpoco 8099 with more context in the substitution hypothesis for 𝑇. (Contributed by SN, 14-Mar-2025.)

Hypotheses

Ref	Expression
fmpocos.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)
fmpocos.2	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))
fmpocos.3	⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))
fmpocos.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇)

Assertion

Ref	Expression
fmpocos	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑧,𝐶,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑧,𝑅   𝑧,𝑇

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑧)   𝑅(𝑥,𝑦)   𝑆(𝑧)   𝑇(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem fmpocos

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmpocos.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)
2	1	ralrimivva 3188	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑅 ∈ 𝐶)
3		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅)
4	3	fmpo 8072	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑅 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
5	2, 4	sylib 218	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
6		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑢𝑅
7		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑣𝑅
8		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
9		nfcsb1v 3903	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝑅
10	8, 9	nfcsbw 3905	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅
11		nfcsb1v 3903	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅
12		csbeq1a 3893	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → 𝑅 = ⦋𝑢 / 𝑥⦌𝑅)
13		csbeq1a 3893	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ⦋𝑢 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
14	12, 13	sylan9eq 2791	. . . . . . 7 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
15	6, 7, 10, 11, 14	cbvmpo 7506	. . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
16		vex 3468	. . . . . . . . 9 ⊢ 𝑢 ∈ V
17		vex 3468	. . . . . . . . 9 ⊢ 𝑣 ∈ V
18	16, 17	op2ndd 8004	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑤) = 𝑣)
19	16, 17	op1std 8003	. . . . . . . . 9 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑤) = 𝑢)
20	19	csbeq1d 3883	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑢 / 𝑥⦌𝑅)
21	18, 20	csbeq12dv 3888	. . . . . . 7 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
22	21	mpompt 7526	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅)
23	15, 22	eqtr4i 2762	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅)
24	23	fmpt 7105	. . . 4 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅):(𝐴 × 𝐵)⟶𝐶)
25	5, 24	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝐴 × 𝐵)⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 ∈ 𝐶)
26		fmpocos.2	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))
27	26, 23	eqtrdi 2787	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅))
28		fmpocos.3	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))
29	25, 27, 28	fmptcos 7126	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆))
30	21	csbeq1d 3883	. . . . 5 ⊢ (𝑤 = ⟨𝑢, 𝑣⟩ → ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆 = ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
31	30	mpompt 7526	. . . 4 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
32		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑢⦋𝑅 / 𝑧⦌𝑆
33		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑣⦋𝑅 / 𝑧⦌𝑆
34		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑆
35	10, 34	nfcsbw 3905	. . . . 5 ⊢ Ⅎ𝑥⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆
36		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝑆
37	11, 36	nfcsbw 3905	. . . . 5 ⊢ Ⅎ𝑦⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆
38	14	csbeq1d 3883	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⦋𝑅 / 𝑧⦌𝑆 = ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
39	32, 33, 35, 37, 38	cbvmpo 7506	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆) = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐵 ↦ ⦋⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝑅 / 𝑧⦌𝑆)
40	31, 39	eqtr4i 2762	. . 3 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆)
41		fmpocos.4	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇)
42	41	3impb 1114	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇)
43	42	mpoeq3dva 7489	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⦋𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
44	40, 43	eqtrid 2783	. 2 ⊢ (𝜑 → (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋⦋(2nd ‘𝑤) / 𝑦⦌⦋(1st ‘𝑤) / 𝑥⦌𝑅 / 𝑧⦌𝑆) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
45	29, 44	eqtrd 2771	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ⦋csb 3879  ⟨cop 4612   ↦ cmpt 5206   × cxp 5657   ∘ ccom 5663  ⟶wf 6532  ‘cfv 6536   ∈ cmpo 7412  1^st c1st 7991  2^nd c2nd 7992

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994

This theorem is referenced by:  evlselv  42577