Theorem fvmpopr2d 6112

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 5977	. . 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 1023	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
4		fvmpopr2d.2	. . . . 5 ⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)
5	4	3ad2ant1 1023	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑃 = ⟨𝑎, 𝑏⟩)
6	3, 5	fveq12d 5610	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)‘⟨𝑎, 𝑏⟩))
7	1, 6	eqtr4id 2261	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝐹‘𝑃))
8		nfcv 2352	. . . . 5 ⊢ Ⅎ𝑐𝐶
9		nfcv 2352	. . . . 5 ⊢ Ⅎ𝑑𝐶
10		nfcv 2352	. . . . . 6 ⊢ Ⅎ𝑎𝑑
11		nfcsb1v 3137	. . . . . 6 ⊢ Ⅎ𝑎⦋𝑐 / 𝑎⦌𝐶
12	10, 11	nfcsbw 3141	. . . . 5 ⊢ Ⅎ𝑎⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
13		nfcsb1v 3137	. . . . 5 ⊢ Ⅎ𝑏⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
14		csbeq1a 3113	. . . . . 6 ⊢ (𝑎 = 𝑐 → 𝐶 = ⦋𝑐 / 𝑎⦌𝐶)
15		csbeq1a 3113	. . . . . 6 ⊢ (𝑏 = 𝑑 → ⦋𝑐 / 𝑎⦌𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
16	14, 15	sylan9eq 2262	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
17	8, 9, 12, 13, 16	cbvmpo 6054	. . . 4 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
18	17	oveqi 5987	. . 3 ⊢ (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏)
19		eqidd 2210	. . . 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 1732	. . . . . . . 8 ⊢ (𝑎 = 𝑐 ↔ 𝑐 = 𝑎)
21		equcom 1732	. . . . . . . 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 2215	. . . . 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 1003	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐴)
27		simp3 1004	. . . 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 6103	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏) = 𝐶)
30	18, 29	eqtrid 2254	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
31	7, 30	eqtr3d 2244	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 983   = wceq 1375   ∈ wcel 2180  ⦋csb 3104  ⟨cop 3649  ‘cfv 5294  (class class class)co 5974   ∈ cmpo 5976

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-iota 5254  df-fun 5296  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979

This theorem is referenced by:  mpomulcn  15205