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Theorem fvmptss 7016
Description: If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping, even if 𝐷 is not in the base set 𝐴. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
Ref Expression
mptrcl.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptss (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mptrcl.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
21dmmptss 6247 . . . 4 dom 𝐹𝐴
32sseli 3972 . . 3 (𝐷 ∈ dom 𝐹𝐷𝐴)
4 fveq2 6896 . . . . . . 7 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
54sseq1d 4008 . . . . . 6 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
65imbi2d 339 . . . . 5 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
7 nfcv 2891 . . . . . 6 𝑥𝑦
8 nfra1 3271 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
9 nfmpt1 5257 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
101, 9nfcxfr 2889 . . . . . . . . 9 𝑥𝐹
1110, 7nffv 6906 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2891 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3969 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
148, 13nfim 1891 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
15 fveq2 6896 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1615sseq1d 4008 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1716imbi2d 339 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
181dmmpt 6246 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
1918reqabi 3441 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
201fvmpt2 7015 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
21 eqimss 4035 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2220, 21syl 17 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2319, 22sylbi 216 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
24 ndmfv 6931 . . . . . . . . . 10 𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ∅)
25 0ss 4398 . . . . . . . . . 10 ∅ ⊆ 𝐵
2624, 25eqsstrdi 4031 . . . . . . . . 9 𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2723, 26pm2.61i 182 . . . . . . . 8 (𝐹𝑥) ⊆ 𝐵
28 rsp 3234 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
2928impcom 406 . . . . . . . 8 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3027, 29sstrid 3988 . . . . . . 7 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3130ex 411 . . . . . 6 (𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
327, 14, 17, 31vtoclgaf 3555 . . . . 5 (𝑦𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
336, 32vtoclga 3556 . . . 4 (𝐷𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3433impcom 406 . . 3 ((∀𝑥𝐴 𝐵𝐶𝐷𝐴) → (𝐹𝐷) ⊆ 𝐶)
353, 34sylan2 591 . 2 ((∀𝑥𝐴 𝐵𝐶𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
36 ndmfv 6931 . . . 4 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
3736adantl 480 . . 3 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) = ∅)
38 0ss 4398 . . 3 ∅ ⊆ 𝐶
3937, 38eqsstrdi 4031 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
4035, 39pm2.61dan 811 1 (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  wss 3944  c0 4322  cmpt 5232  dom cdm 5678  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557
This theorem is referenced by:  relmptopab  7671  ovmptss  8098
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