Theorem fvmptss2 5726

Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypotheses

Ref	Expression
fvmptn.1	⊢ (x = D → B = C)
fvmptn.2	⊢ F = (x ∈ A ↦ B)

Assertion

Ref	Expression
fvmptss2	⊢ (F ‘D) ⊆ C

Distinct variable groups:   x,A   x,C   x,D

Allowed substitution hints:   B(x)   F(x)

Proof of Theorem fvmptss2

Step	Hyp	Ref	Expression
1		fvmptn.1	. . . . 5 ⊢ (x = D → B = C)
2	1	eleq1d 2419	. . . 4 ⊢ (x = D → (B ∈ V ↔ C ∈ V))
3		fvmptn.2	. . . . 5 ⊢ F = (x ∈ A ↦ B)
4	3	dmmpt 5684	. . . 4 ⊢ dom F = {x ∈ A ∣ B ∈ V}
5	2, 4	elrab2 2997	. . 3 ⊢ (D ∈ dom F ↔ (D ∈ A ∧ C ∈ V))
6	1, 3	fvmptg 5699	. . . 4 ⊢ ((D ∈ A ∧ C ∈ V) → (F ‘D) = C)
7		eqimss 3324	. . . 4 ⊢ ((F ‘D) = C → (F ‘D) ⊆ C)
8	6, 7	syl 15	. . 3 ⊢ ((D ∈ A ∧ C ∈ V) → (F ‘D) ⊆ C)
9	5, 8	sylbi 187	. 2 ⊢ (D ∈ dom F → (F ‘D) ⊆ C)
10		ndmfv 5350	. . 3 ⊢ (¬ D ∈ dom F → (F ‘D) = ∅)
11		0ss 3580	. . . 4 ⊢ ∅ ⊆ C
12	11	a1i 10	. . 3 ⊢ (¬ D ∈ dom F → ∅ ⊆ C)
13	10, 12	eqsstrd 3306	. 2 ⊢ (¬ D ∈ dom F → (F ‘D) ⊆ C)
14	9, 13	pm2.61i 156	1 ⊢ (F ‘D) ⊆ C

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  ∅c0 3551  dom cdm 4773   ‘cfv 4782   ↦ cmpt 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-mpt 5653

This theorem is referenced by: (None)