Theorem dmmpt2ssx 7235

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

Hypothesis

Ref	Expression
fmpt2x.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
dmmpt2ssx	⊢ 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 dmmpt2ssx

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

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑢𝐵
2		nfcsb1v 3549	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
3		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcsb1v 3549	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
6		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑦𝑢
7		nfcsb1v 3549	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
8	6, 7	nfcsb 3551	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
9		csbeq1a 3542	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
10		csbeq1a 3542	. . . . . 6 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
11		csbeq1a 3542	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
12	10, 11	sylan9eqr 2678	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpt2x 6733	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
14		fmpt2x.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
15		vex 3203	. . . . . . . 8 ⊢ 𝑢 ∈ V
16		vex 3203	. . . . . . . 8 ⊢ 𝑣 ∈ V
17	15, 16	op1std 7178	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑡) = 𝑢)
18	17	csbeq1d 3540	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
19	15, 16	op2ndd 7179	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑡) = 𝑣)
20	19	csbeq1d 3540	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
21	20	csbeq2dv 3992	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
22	18, 21	eqtrd 2656	. . . . 5 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
23	22	mpt2mptx 6751	. . . 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 2654	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
25	24	dmmptss 5631	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
26		nfcv 2764	. . 3 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
27		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥{𝑢}
28	27, 2	nfxp 5142	. . 3 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
29		sneq 4187	. . . 4 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
30	29, 9	xpeq12d 5140	. . 3 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
31	26, 28, 30	cbviun 4557	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
32	25, 31	sseqtr4i 3638	1 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

Syntax hints:   = wceq 1483  ⦋csb 3533   ⊆ wss 3574  {csn 4177  ⟨cop 4183  ∪ ciun 4520   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ‘cfv 5888   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mpt2exxg  7244  mpt2xeldm  7337  mpt2xopn0yelv  7339  mpt2xopxnop0  7341  dmcoass  16716  ply1frcl  19683  dvbsss  23666  perfdvf  23667