Theorem fimaproj 8154

Description: Image of a cartesian product for a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)

Hypotheses

Ref	Expression
fvproj.h	⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
fimaproj.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
fimaproj.g	⊢ (𝜑 → 𝐺 Fn 𝐵)
fimaproj.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
fimaproj.y	⊢ (𝜑 → 𝑌 ⊆ 𝐵)

Assertion

Ref	Expression
fimaproj	⊢ (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑋(𝑥, 𝑦)   𝑌(𝑥, 𝑦)

Proof of Theorem fimaproj

Dummy variables 𝑎 𝑏 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5431	. . . . 5 ⊢ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ V
2		fvproj.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
3		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
4		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5	3, 4	op1std 8011	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
6	5	fveq2d 6889	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
7	3, 4	op2ndd 8012	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
8	7	fveq2d 6889	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2nd ‘𝑧)) = (𝐺‘𝑦))
9	6, 8	opeq12d 4847	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
10	9	mpompt 7534	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
11	2, 10	eqtr4i 2786	. . . . 5 ⊢ 𝐻 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩)
12	1, 11	fnmpti 6682	. . . 4 ⊢ 𝐻 Fn (𝐴 × 𝐵)
13		fimaproj.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
14		fimaproj.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝐵)
15		xpss12 5665	. . . . 5 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
16	13, 14, 15	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
17		fvelimab 6957	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐵) ∧ (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵)) → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐))
18	12, 16, 17	sylancr 599	. . 3 ⊢ (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐))
19		simp-4r 796	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑎 ∈ 𝑋)
20		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑏 ∈ 𝑌)
21		opelxpi 5687	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
22	19, 20, 21	syl2anc 596	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
23		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐹‘𝑎) = (1st ‘𝑐))
24		simpr 490	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐺‘𝑏) = (2nd ‘𝑐))
25	23, 24	opeq12d 4847	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩ = ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)
26	13	ad5antr 747	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑋 ⊆ 𝐴)
27	26, 19	sseldd 3909	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑎 ∈ 𝐴)
28	14	ad5antr 747	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑌 ⊆ 𝐵)
29	28, 20	sseldd 3909	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑏 ∈ 𝐵)
30	2, 27, 29	fvproj 8153	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩)
31		1st2nd2 8040	. . . . . . . . 9 ⊢ (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) → 𝑐 = ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)
32	31	ad5antlr 748	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑐 = ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)
33	25, 30, 32	3eqtr4d 2805	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐)
34		fveqeq2 6894	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → ((𝐻‘𝑧) = 𝑐 ↔ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐))
35	34	rspcev 3576	. . . . . . 7 ⊢ ((⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌) ∧ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
36	22, 33, 35	syl2anc 596	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
37		fimaproj.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝐵)
38	37	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → 𝐺 Fn 𝐵)
39		fnfun 6639	. . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
40	38, 39	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → Fun 𝐺)
41		xp2nd 8034	. . . . . . . 8 ⊢ (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) → (2nd ‘𝑐) ∈ (𝐺 “ 𝑌))
42	41	ad3antlr 744	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → (2nd ‘𝑐) ∈ (𝐺 “ 𝑌))
43		fvelima 6950	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ (2nd ‘𝑐) ∈ (𝐺 “ 𝑌)) → ∃𝑏 ∈ 𝑌 (𝐺‘𝑏) = (2nd ‘𝑐))
44	40, 42, 43	syl2anc 596	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → ∃𝑏 ∈ 𝑌 (𝐺‘𝑏) = (2nd ‘𝑐))
45	36, 44	r19.29a 3170	. . . . 5 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
46		fimaproj.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝐴)
47	46	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → 𝐹 Fn 𝐴)
48		fnfun 6639	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
49	47, 48	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → Fun 𝐹)
50		xp1st 8033	. . . . . . 7 ⊢ (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) → (1st ‘𝑐) ∈ (𝐹 “ 𝑋))
51	50	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → (1st ‘𝑐) ∈ (𝐹 “ 𝑋))
52		fvelima 6950	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (1st ‘𝑐) ∈ (𝐹 “ 𝑋)) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = (1st ‘𝑐))
53	49, 51, 52	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = (1st ‘𝑐))
54	45, 53	r19.29a 3170	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
55		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐻‘𝑧) = 𝑐)
56	16	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
57		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑧 ∈ (𝑋 × 𝑌))
58	56, 57	sseldd 3909	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑧 ∈ (𝐴 × 𝐵))
59	11	fvmpt2 7005	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ V) → (𝐻‘𝑧) = ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩)
60	58, 1, 59	sylancl 598	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐻‘𝑧) = ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩)
61	46	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝐹 Fn 𝐴)
62	13	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑋 ⊆ 𝐴)
63		xp1st 8033	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (1st ‘𝑧) ∈ 𝑋)
64	57, 63	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (1st ‘𝑧) ∈ 𝑋)
65		fnfvima 7239	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ∧ (1st ‘𝑧) ∈ 𝑋) → (𝐹‘(1st ‘𝑧)) ∈ (𝐹 “ 𝑋))
66	61, 62, 64, 65	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐹‘(1st ‘𝑧)) ∈ (𝐹 “ 𝑋))
67	37	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝐺 Fn 𝐵)
68	14	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑌 ⊆ 𝐵)
69		xp2nd 8034	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2nd ‘𝑧) ∈ 𝑌)
70	57, 69	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (2nd ‘𝑧) ∈ 𝑌)
71		fnfvima 7239	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑌 ⊆ 𝐵 ∧ (2nd ‘𝑧) ∈ 𝑌) → (𝐺‘(2nd ‘𝑧)) ∈ (𝐺 “ 𝑌))
72	67, 68, 70, 71	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐺‘(2nd ‘𝑧)) ∈ (𝐺 “ 𝑌))
73		opelxpi 5687	. . . . . . . 8 ⊢ (((𝐹‘(1st ‘𝑧)) ∈ (𝐹 “ 𝑋) ∧ (𝐺‘(2nd ‘𝑧)) ∈ (𝐺 “ 𝑌)) → ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
74	66, 72, 73	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
75	60, 74	eqeltrd 2860	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐻‘𝑧) ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
76	55, 75	eqeltrrd 2861	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
77	76	r19.29an 3166	. . . 4 ⊢ ((𝜑 ∧ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐) → 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
78	54, 77	impbida 813	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐))
79	18, 78	bitr4d 285	. 2 ⊢ (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))))
80	79	eqrdv 2758	1 ⊢ (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ⟨cop 4834   ↦ cmpt 5185   × cxp 5648   “ cima 5653  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002

This theorem is used by:  txomap  34512