MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmptss Structured version   Visualization version   GIF version

Theorem ovmptss 7413
Description: If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
ovmptss.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
ovmptss (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem ovmptss
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovmptss.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 mpt2mptsx 7387 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
31, 2eqtri 2793 . . 3 𝐹 = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
43fvmptss 6436 . 2 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 → (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
5 vex 3354 . . . . . . . 8 𝑢 ∈ V
6 vex 3354 . . . . . . . 8 𝑣 ∈ V
75, 6op1std 7329 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
87csbeq1d 3689 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
95, 6op2ndd 7330 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
109csbeq1d 3689 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
1110csbeq2dv 4137 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
128, 11eqtrd 2805 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1312sseq1d 3781 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
1413raliunxp 5399 . . 3 (∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
15 nfcv 2913 . . . . 5 𝑢({𝑥} × 𝐵)
16 nfcv 2913 . . . . . 6 𝑥{𝑢}
17 nfcsb1v 3698 . . . . . 6 𝑥𝑢 / 𝑥𝐵
1816, 17nfxp 5282 . . . . 5 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
19 sneq 4327 . . . . . 6 (𝑥 = 𝑢 → {𝑥} = {𝑢})
20 csbeq1a 3691 . . . . . 6 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
2119, 20xpeq12d 5280 . . . . 5 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
2215, 18, 21cbviun 4692 . . . 4 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
2322raleqi 3291 . . 3 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
24 nfv 1995 . . . 4 𝑢𝑦𝐵 𝐶𝑋
25 nfcsb1v 3698 . . . . . 6 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
26 nfcv 2913 . . . . . 6 𝑥𝑋
2725, 26nfss 3745 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
2817, 27nfral 3094 . . . 4 𝑥𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
29 nfv 1995 . . . . . 6 𝑣 𝐶𝑋
30 nfcsb1v 3698 . . . . . . 7 𝑦𝑣 / 𝑦𝐶
31 nfcv 2913 . . . . . . 7 𝑦𝑋
3230, 31nfss 3745 . . . . . 6 𝑦𝑣 / 𝑦𝐶𝑋
33 csbeq1a 3691 . . . . . . 7 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
3433sseq1d 3781 . . . . . 6 (𝑦 = 𝑣 → (𝐶𝑋𝑣 / 𝑦𝐶𝑋))
3529, 32, 34cbvral 3316 . . . . 5 (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋)
36 csbeq1a 3691 . . . . . . 7 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
3736sseq1d 3781 . . . . . 6 (𝑥 = 𝑢 → (𝑣 / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3820, 37raleqbidv 3301 . . . . 5 (𝑥 = 𝑢 → (∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3935, 38syl5bb 272 . . . 4 (𝑥 = 𝑢 → (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
4024, 28, 39cbvral 3316 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
4114, 23, 403bitr4ri 293 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
42 df-ov 6799 . . 3 (𝐸𝐹𝐺) = (𝐹‘⟨𝐸, 𝐺⟩)
4342sseq1i 3778 . 2 ((𝐸𝐹𝐺) ⊆ 𝑋 ↔ (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
444, 41, 433imtr4i 281 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wral 3061  csb 3682  wss 3723  {csn 4317  cop 4323   ciun 4655  cmpt 4864   × cxp 5248  cfv 6030  (class class class)co 6796  cmpt2 6798  1st c1st 7317  2nd c2nd 7318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320
This theorem is referenced by:  relmpt2opab  7414  relxpchom  17029  reldv  23854
  Copyright terms: Public domain W3C validator