Theorem fnmpoovd 8088

Description: A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)

Hypotheses

Ref	Expression
fnmpoovd.m	⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))
fnmpoovd.s	⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
fnmpoovd.d	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
fnmpoovd.c	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fnmpoovd	⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑖,𝑗   𝐵,𝑎,𝑏,𝑖,𝑗   𝐶,𝑖,𝑗   𝐷,𝑎,𝑏   𝑖,𝑀,𝑗   𝜑,𝑎,𝑏,𝑖,𝑗

Allowed substitution hints:   𝐶(𝑎,𝑏)   𝐷(𝑖,𝑗)   𝑈(𝑖,𝑗,𝑎,𝑏)   𝑀(𝑎,𝑏)   𝑉(𝑖,𝑗,𝑎,𝑏)

Proof of Theorem fnmpoovd

Step	Hyp	Ref	Expression
1		fnmpoovd.m	. . 3 ⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))
2		fnmpoovd.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)
3	2	3expb 1117	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
4	3	ralrimivva 3191	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉)
5		eqid 2725	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
6	5	fnmpo 8069	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
7	4, 6	syl 17	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
8		eqfnov2 7546	. . 3 ⊢ ((𝑀 Fn (𝐴 × 𝐵) ∧ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵)) → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
9	1, 7, 8	syl2anc 582	. 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
10		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑎𝐷
11		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑏𝐷
12		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑖𝐶
13		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑗𝐶
14		fnmpoovd.s	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
15	10, 11, 12, 13, 14	cbvmpo 7509	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
16	15	eqcomi 2734	. . . . . 6 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷))
18	17	oveqd 7431	. . . 4 ⊢ (𝜑 → (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗))
19	18	eqeq2d 2736	. . 3 ⊢ (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
20	19	2ralbidv 3209	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
21		simprl 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑖 ∈ 𝐴)
22		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑗 ∈ 𝐵)
23		fnmpoovd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
24	23	3expb 1117	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝐷 ∈ 𝑈)
25		eqid 2725	. . . . . 6 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
26	25	ovmpt4g 7563	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵 ∧ 𝐷 ∈ 𝑈) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
27	21, 22, 24, 26	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
28	27	eqeq2d 2736	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29	28	2ralbidva 3207	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))
30	9, 20, 29	3bitrd 304	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3051   × cxp 5668   Fn wfn 6536  (class class class)co 7414   ∈ cmpo 7416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pr 5421  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990

This theorem is referenced by:  mpofrlmd  21713  fedgmullem2  33357