Theorem fvmptss2 5725

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 5683	. . . 4 ⊢ dom F = {x ∈ A ∣ B ∈ V}
5	2, 4	elrab2 2996	. . 3 ⊢ (D ∈ dom F ↔ (D ∈ A ∧ C ∈ V))
6	1, 3	fvmptg 5698	. . . 4 ⊢ ((D ∈ A ∧ C ∈ V) → (F ‘D) = C)
7		eqimss 3323	. . . 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 5349	. . 3 ⊢ (¬ D ∈ dom F → (F ‘D) = ∅)
11		0ss 3579	. . . 4 ⊢ ∅ ⊆ C
12	11	a1i 10	. . 3 ⊢ (¬ D ∈ dom F → ∅ ⊆ C)
13	10, 12	eqsstrd 3305	. 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 2859   ⊆ wss 3257  ∅c0 3550  dom cdm 4772   ‘cfv 4781   ↦ cmpt 5651

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-mpt 5652

This theorem is referenced by: (None)