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Theorem fvmptssdm 5621
Description: If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
Hypothesis
Ref Expression
fvmpt2.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptssdm ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptssdm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 5534 . . . . . 6 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
21sseq1d 3199 . . . . 5 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
32imbi2d 230 . . . 4 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
4 nfrab1 2670 . . . . . . 7 𝑥{𝑥𝐴𝐵 ∈ V}
54nfcri 2326 . . . . . 6 𝑥 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}
6 nfra1 2521 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
7 fvmpt2.1 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
8 nfmpt1 4111 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
97, 8nfcxfr 2329 . . . . . . . . 9 𝑥𝐹
10 nfcv 2332 . . . . . . . . 9 𝑥𝑦
119, 10nffv 5544 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2332 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3163 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
146, 13nfim 1583 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
155, 14nfim 1583 . . . . 5 𝑥(𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
16 eleq1 2252 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}))
17 fveq2 5534 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1817sseq1d 3199 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1918imbi2d 230 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
2016, 19imbi12d 234 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶)) ↔ (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))))
217dmmpt 5142 . . . . . . 7 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2221eleq2i 2256 . . . . . 6 (𝑥 ∈ dom 𝐹𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
2321rabeq2i 2749 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
247fvmpt2 5620 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
25 eqimss 3224 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2624, 25syl 14 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2723, 26sylbi 121 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2827adantr 276 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐵)
297dmmptss 5143 . . . . . . . . . 10 dom 𝐹𝐴
3029sseli 3166 . . . . . . . . 9 (𝑥 ∈ dom 𝐹𝑥𝐴)
31 rsp 2537 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
3230, 31mpan9 281 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3328, 32sstrd 3180 . . . . . . 7 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3433ex 115 . . . . . 6 (𝑥 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3522, 34sylbir 135 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3615, 20, 35chvar 1768 . . . 4 (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
373, 36vtoclga 2818 . . 3 (𝐷 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3837, 21eleq2s 2284 . 2 (𝐷 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3938imp 124 1 ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  {crab 2472  Vcvv 2752  wss 3144  cmpt 4079  dom cdm 4644  cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fv 5243
This theorem is referenced by: (None)
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