Theorem fnmpoovd 6385

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 1230	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
4	3	ralrimivva 2613	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉)
5		eqid 2230	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
6	5	fnmpo 6372	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
7	4, 6	syl 14	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
8		eqfnov2 6134	. . 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 411	. 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
10		nfcv 2373	. . . . . . . 8 ⊢ Ⅎ𝑎𝐷
11		nfcv 2373	. . . . . . . 8 ⊢ Ⅎ𝑏𝐷
12		nfcv 2373	. . . . . . . 8 ⊢ Ⅎ𝑖𝐶
13		nfcv 2373	. . . . . . . 8 ⊢ Ⅎ𝑗𝐶
14		fnmpoovd.s	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
15	10, 11, 12, 13, 14	cbvmpo 6105	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
16	15	eqcomi 2234	. . . . . 6 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
17	16	a1i 9	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷))
18	17	oveqd 6040	. . . 4 ⊢ (𝜑 → (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗))
19	18	eqeq2d 2242	. . 3 ⊢ (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
20	19	2ralbidv 2555	. 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 531	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑖 ∈ 𝐴)
22		simprr 533	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑗 ∈ 𝐵)
23		fnmpoovd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
24	23	3expb 1230	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝐷 ∈ 𝑈)
25		eqid 2230	. . . . . 6 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
26	25	ovmpt4g 6149	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵 ∧ 𝐷 ∈ 𝑈) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
27	21, 22, 24, 26	syl3anc 1273	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
28	27	eqeq2d 2242	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29	28	2ralbidva 2553	. 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 214	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2201  ∀wral 2509   × cxp 4725   Fn wfn 5323  (class class class)co 6023   ∈ cmpo 6025

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309

This theorem is referenced by: (None)