Theorem fvmpopr2d 6147

Description: Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)

Hypotheses

Ref	Expression
fvmpopr2d.1	⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
fvmpopr2d.2	⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)
fvmpopr2d.3	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fvmpopr2d	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)

Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝑃(𝑎,𝑏)   𝐹(𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem fvmpopr2d

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6010	. . 3 ⊢ (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)‘⟨𝑎, 𝑏⟩)
2		fvmpopr2d.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
3	2	3ad2ant1 1042	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
4		fvmpopr2d.2	. . . . 5 ⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)
5	4	3ad2ant1 1042	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑃 = ⟨𝑎, 𝑏⟩)
6	3, 5	fveq12d 5636	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)‘⟨𝑎, 𝑏⟩))
7	1, 6	eqtr4id 2281	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝐹‘𝑃))
8		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑐𝐶
9		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑑𝐶
10		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑎𝑑
11		nfcsb1v 3157	. . . . . 6 ⊢ Ⅎ𝑎⦋𝑐 / 𝑎⦌𝐶
12	10, 11	nfcsbw 3161	. . . . 5 ⊢ Ⅎ𝑎⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
13		nfcsb1v 3157	. . . . 5 ⊢ Ⅎ𝑏⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
14		csbeq1a 3133	. . . . . 6 ⊢ (𝑎 = 𝑐 → 𝐶 = ⦋𝑐 / 𝑎⦌𝐶)
15		csbeq1a 3133	. . . . . 6 ⊢ (𝑏 = 𝑑 → ⦋𝑐 / 𝑎⦌𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
16	14, 15	sylan9eq 2282	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
17	8, 9, 12, 13, 16	cbvmpo 6089	. . . 4 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
18	17	oveqi 6020	. . 3 ⊢ (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏)
19		eqidd 2230	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶) = (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶))
20		equcom 1752	. . . . . . . 8 ⊢ (𝑎 = 𝑐 ↔ 𝑐 = 𝑎)
21		equcom 1752	. . . . . . . 8 ⊢ (𝑏 = 𝑑 ↔ 𝑑 = 𝑏)
22	20, 21	anbi12i 460	. . . . . . 7 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) ↔ (𝑐 = 𝑎 ∧ 𝑑 = 𝑏))
23	22, 16	sylbir 135	. . . . . 6 ⊢ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
24	23	eqcomd 2235	. . . . 5 ⊢ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) → ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶 = 𝐶)
25	24	adantl 277	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 = 𝑎 ∧ 𝑑 = 𝑏)) → ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶 = 𝐶)
26		simp2 1022	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐴)
27		simp3 1023	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
28		fvmpopr2d.3	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)
29	19, 25, 26, 27, 28	ovmpod 6138	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏) = 𝐶)
30	18, 29	eqtrid 2274	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
31	7, 30	eqtr3d 2264	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ⦋csb 3124  ⟨cop 3669  ‘cfv 5318  (class class class)co 6007   ∈ cmpo 6009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012

This theorem is referenced by:  mpomulcn  15248