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Theorem fvmptss 6331
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 5669 . . . 4 dom 𝐹𝐴
32sseli 3632 . . 3 (𝐷 ∈ dom 𝐹𝐷𝐴)
4 fveq2 6229 . . . . . . 7 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
54sseq1d 3665 . . . . . 6 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
65imbi2d 329 . . . . 5 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
7 nfcv 2793 . . . . . 6 𝑥𝑦
8 nfra1 2970 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
9 nfmpt1 4780 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
101, 9nfcxfr 2791 . . . . . . . . 9 𝑥𝐹
1110, 7nffv 6236 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2793 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3629 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
148, 13nfim 1865 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
15 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1615sseq1d 3665 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1716imbi2d 329 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
181dmmpt 5668 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
1918rabeq2i 3228 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
201fvmpt2 6330 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
21 eqimss 3690 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2220, 21syl 17 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2319, 22sylbi 207 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
24 ndmfv 6256 . . . . . . . . . 10 𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ∅)
25 0ss 4005 . . . . . . . . . 10 ∅ ⊆ 𝐵
2624, 25syl6eqss 3688 . . . . . . . . 9 𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2723, 26pm2.61i 176 . . . . . . . 8 (𝐹𝑥) ⊆ 𝐵
28 rsp 2958 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
2928impcom 445 . . . . . . . 8 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3027, 29syl5ss 3647 . . . . . . 7 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3130ex 449 . . . . . 6 (𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
327, 14, 17, 31vtoclgaf 3302 . . . . 5 (𝑦𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
336, 32vtoclga 3303 . . . 4 (𝐷𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3433impcom 445 . . 3 ((∀𝑥𝐴 𝐵𝐶𝐷𝐴) → (𝐹𝐷) ⊆ 𝐶)
353, 34sylan2 490 . 2 ((∀𝑥𝐴 𝐵𝐶𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
36 ndmfv 6256 . . . 4 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
3736adantl 481 . . 3 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) = ∅)
38 0ss 4005 . . 3 ∅ ⊆ 𝐶
3937, 38syl6eqss 3688 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
4035, 39pm2.61dan 849 1 (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948  cmpt 4762  dom cdm 5143  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  relmptopab  6925  ovmptss  7303
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