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Theorem fparlem3 7882

Description: Lemma for fpar 7884. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Assertion**
Ref	Expression
fparlem3	⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐹

Proof of Theorem fparlem3 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coiun 6120	. 2 ⊢ (^◡(1^st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = ∪ 𝑥 ∈ 𝐴 (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2		inss1 4143	. . . . 5 ⊢ (dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ dom 𝐹
3		fndm 6481	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3953	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ 𝐴)
5		dfco2a 6110	. . . 4 ⊢ ((dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ 𝐴 → (𝐹 ∘ (1^st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
6	4, 5	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ (1^st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
7	6	coeq2d 5731	. 2 ⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = (^◡(1^st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8		inss1 4143	. . . . . . . . 9 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ dom ({(𝐹‘𝑥)} × ({𝑥} × V))
9		dmxpss 6034	. . . . . . . . 9 ⊢ dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ⊆ {(𝐹‘𝑥)}
10	8, 9	sstri 3910	. . . . . . . 8 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ {(𝐹‘𝑥)}
11		dfco2a 6110	. . . . . . . 8 ⊢ ((dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ {(𝐹‘𝑥)} → (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}))
13		fvex 6730	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
14		fparlem1 7880	. . . . . . . . . 10 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15		sneq 4551	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → {𝑦} = {(𝐹‘𝑥)})
16	15	xpeq1d 5580	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ({𝑦} × V) = ({(𝐹‘𝑥)} × V))
17	14, 16	eqtrid 2789	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (^◡(1^st ↾ (V × V)) “ {𝑦}) = ({(𝐹‘𝑥)} × V))
18	15	imaeq2d 5929	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}))
19		df-ima 5564	. . . . . . . . . . 11 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)})
20		ssid 3923	. . . . . . . . . . . . . 14 ⊢ {(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)}
21		xpssres 5888	. . . . . . . . . . . . . 14 ⊢ ({(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)} → (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({(𝐹‘𝑥)} × ({𝑥} × V)))
22	20, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({(𝐹‘𝑥)} × ({𝑥} × V))
23	22	rneqi 5806	. . . . . . . . . . . 12 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ran ({(𝐹‘𝑥)} × ({𝑥} × V))
24	13	snnz 4692	. . . . . . . . . . . . 13 ⊢ {(𝐹‘𝑥)} ≠ ∅
25		rnxp 6033	. . . . . . . . . . . . 13 ⊢ ({(𝐹‘𝑥)} ≠ ∅ → ran ({(𝐹‘𝑥)} × ({𝑥} × V)) = ({𝑥} × V))
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran ({(𝐹‘𝑥)} × ({𝑥} × V)) = ({𝑥} × V)
27	23, 26	eqtri 2765	. . . . . . . . . . 11 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({𝑥} × V)
28	19, 27	eqtri 2765	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ({𝑥} × V)
29	18, 28	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
30	17, 29	xpeq12d 5582	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V)))
31	13, 30	iunxsn 4999	. . . . . . 7 ⊢ ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
32	12, 31	eqtri 2765	. . . . . 6 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
33	32	cnveqi 5743	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ^◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V))
34		cnvco 5754	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V)))
35		cnvxp 6020	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
36	33, 34, 35	3eqtr3i 2773	. . . 4 ⊢ (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
37		fparlem1 7880	. . . . . . . . 9 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
38	37	xpeq2i 5578	. . . . . . . 8 ⊢ ({(𝐹‘𝑥)} × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ({(𝐹‘𝑥)} × ({𝑥} × V))
39		fnsnfv 6790	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
40	39	xpeq1d 5580	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
41	38, 40	eqtr3id 2792	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
42	41	cnveqd 5744	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ^◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ^◡((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
43		cnvxp 6020	. . . . . 6 ⊢ ^◡((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
44	42, 43	eqtrdi 2794	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ^◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
45	44	coeq2d 5731	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
46	36, 45	eqtr3id 2792	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
47	46	iuneq2dv 4928	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = ∪ 𝑥 ∈ 𝐴 (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
48	1, 7, 47	3eqtr4a 2804	1 ⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 ∅c0 4237 {csn 4541 ∪ ciun 4904 × cxp 5549 ^◡ccnv 5550 dom cdm 5551 ran crn 5552 ↾ cres 5553 “ cima 5554 ∘ ccom 5555 Fn wfn 6375 ‘cfv 6380 1^st c1st 7759

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-1st 7761 df-2nd 7762

This theorem is referenced by: fpar 7884

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