Theorem fvmptss2 7018

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 2848	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3		fvmptn.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmpt 6243	. . . 4 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	2, 4	elrab2 3655	. . 3 ⊢ (𝐷 ∈ dom 𝐹 ↔ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V))
6	1, 3	fvmptg 6989	. . . 4 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) = 𝐶)
7		eqimss 3996	. . . 4 ⊢ ((𝐹‘𝐷) = 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)
8	6, 7	syl 18	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) ⊆ 𝐶)
9	5, 8	sylbi 220	. 2 ⊢ (𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶)
10		ndmfv 6915	. . 3 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅)
11		0ss 4358	. . 3 ⊢ ∅ ⊆ 𝐶
12	10, 11	eqsstrdi 3982	. 2 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶)
13	9, 12	pm2.61i 184	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3906  ∅c0 4287   ↦ cmpt 5193  dom cdm 5663  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is referenced by:  cvmsi  35735