Theorem dmmpossx 6343

Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypothesis

Ref	Expression
fmpox.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
dmmpossx	⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpossx

Dummy variables 𝑢 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑢𝐵
2		nfcsb1v 3157	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
3		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcsb1v 3157	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
6		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑦𝑢
7		nfcsb1v 3157	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
8	6, 7	nfcsb 3162	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
9		csbeq1a 3133	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
10		csbeq1a 3133	. . . . . 6 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
11		csbeq1a 3133	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
12	10, 11	sylan9eqr 2284	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 6081	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
14		fmpox.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
15		vex 2802	. . . . . . . 8 ⊢ 𝑢 ∈ V
16		vex 2802	. . . . . . . 8 ⊢ 𝑣 ∈ V
17	15, 16	op1std 6292	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑡) = 𝑢)
18	17	csbeq1d 3131	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
19	15, 16	op2ndd 6293	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑡) = 𝑣)
20	19	csbeq1d 3131	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
21	20	csbeq2dv 3150	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
22	18, 21	eqtrd 2262	. . . . 5 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
23	22	mpomptx 6094	. . . 4 ⊢ (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
24	13, 14, 23	3eqtr4i 2260	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
25	24	dmmptss 5224	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
26		nfcv 2372	. . 3 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
27		nfcv 2372	. . . 4 ⊢ Ⅎ𝑥{𝑢}
28	27, 2	nfxp 4745	. . 3 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
29		sneq 3677	. . . 4 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
30	29, 9	xpeq12d 4743	. . 3 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
31	26, 28, 30	cbviun 4001	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
32	25, 31	sseqtrri 3259	1 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

Syntax hints:   = wceq 1395  ⦋csb 3124   ⊆ wss 3197  {csn 3666  ⟨cop 3669  ∪ ciun 3964   ↦ cmpt 4144   × cxp 4716  dom cdm 4718  ‘cfv 5317   ∈ cmpo 6002  1^st c1st 6282  2^nd c2nd 6283

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fv 5325  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285

This theorem is referenced by:  mpoexxg  6354  mpoxopn0yelv  6383