Theorem fvmptss2 7012

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 2846	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3		fvmptn.2	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmpt 6234	. . . 4 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
5	2, 4	elrab2 3649	. . 3 ⊢ (𝐷 ∈ dom 𝐹 ↔ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V))
6	1, 3	fvmptg 6983	. . . 4 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ V) → (𝐹‘𝐷) = 𝐶)
7		eqimss 3989	. . . 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 6909	. . 3 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) = ∅)
11		0ss 4350	. . 3 ⊢ ∅ ⊆ 𝐶
12	10, 11	eqsstrdi 3975	. 2 ⊢ (¬ 𝐷 ∈ dom 𝐹 → (𝐹‘𝐷) ⊆ 𝐶)
13	9, 12	pm2.61i 184	1 ⊢ (𝐹‘𝐷) ⊆ 𝐶

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  ∅c0 4279   ↦ cmpt 5186  dom cdm 5651  ‘cfv 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539

This theorem is used by:  cvmsi  35999