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Theorem ovmptss 7778
 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 mpomptsx 7751 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
31, 2eqtri 2821 . . 3 𝐹 = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
43fvmptss 6762 . 2 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 → (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
5 vex 3444 . . . . . . . 8 𝑢 ∈ V
6 vex 3444 . . . . . . . 8 𝑣 ∈ V
75, 6op1std 7688 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
87csbeq1d 3832 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
95, 6op2ndd 7689 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
109csbeq1d 3832 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
1110csbeq2dv 3835 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
128, 11eqtrd 2833 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1312sseq1d 3946 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
1413raliunxp 5675 . . 3 (∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
15 nfcv 2955 . . . . 5 𝑢({𝑥} × 𝐵)
16 nfcv 2955 . . . . . 6 𝑥{𝑢}
17 nfcsb1v 3852 . . . . . 6 𝑥𝑢 / 𝑥𝐵
1816, 17nfxp 5553 . . . . 5 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
19 sneq 4535 . . . . . 6 (𝑥 = 𝑢 → {𝑥} = {𝑢})
20 csbeq1a 3842 . . . . . 6 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
2119, 20xpeq12d 5551 . . . . 5 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
2215, 18, 21cbviun 4924 . . . 4 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
2322raleqi 3362 . . 3 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
24 nfv 1915 . . . 4 𝑢𝑦𝐵 𝐶𝑋
25 nfcsb1v 3852 . . . . . 6 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
26 nfcv 2955 . . . . . 6 𝑥𝑋
2725, 26nfss 3907 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
2817, 27nfralw 3189 . . . 4 𝑥𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
29 nfv 1915 . . . . . 6 𝑣 𝐶𝑋
30 nfcsb1v 3852 . . . . . . 7 𝑦𝑣 / 𝑦𝐶
31 nfcv 2955 . . . . . . 7 𝑦𝑋
3230, 31nfss 3907 . . . . . 6 𝑦𝑣 / 𝑦𝐶𝑋
33 csbeq1a 3842 . . . . . . 7 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
3433sseq1d 3946 . . . . . 6 (𝑦 = 𝑣 → (𝐶𝑋𝑣 / 𝑦𝐶𝑋))
3529, 32, 34cbvralw 3387 . . . . 5 (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋)
36 csbeq1a 3842 . . . . . . 7 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
3736sseq1d 3946 . . . . . 6 (𝑥 = 𝑢 → (𝑣 / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3820, 37raleqbidv 3354 . . . . 5 (𝑥 = 𝑢 → (∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3935, 38syl5bb 286 . . . 4 (𝑥 = 𝑢 → (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
4024, 28, 39cbvralw 3387 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
4114, 23, 403bitr4ri 307 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
42 df-ov 7143 . . 3 (𝐸𝐹𝐺) = (𝐹‘⟨𝐸, 𝐺⟩)
4342sseq1i 3943 . 2 ((𝐸𝐹𝐺) ⊆ 𝑋 ↔ (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
444, 41, 433imtr4i 295 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∀wral 3106  ⦋csb 3828   ⊆ wss 3881  {csn 4525  ⟨cop 4531  ∪ ciun 4882   ↦ cmpt 5111   × cxp 5518  ‘cfv 6327  (class class class)co 7140   ∈ cmpo 7142  1st c1st 7676  2nd c2nd 7677 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7678  df-2nd 7679 This theorem is referenced by:  relmpoopab  7779  relxpchom  17430  reldv  24487
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