Theorem fvopabf4g 35879

Description: Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
fvopabf4g.1	⊢ 𝐶 ∈ V
fvopabf4g.2	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
fvopabf4g.3	⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵)

Assertion

Ref	Expression
fvopabf4g	⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶)

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

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

Proof of Theorem fvopabf4g

Step	Hyp	Ref	Expression
1		elmapg 8628	. . . 4 ⊢ ((𝑅 ∈ 𝑌 ∧ 𝐷 ∈ 𝑋) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅))
2	1	ancoms 459	. . 3 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅))
3	2	biimp3ar 1469	. 2 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → 𝐴 ∈ (𝑅 ↑m 𝐷))
4		fvopabf4g.2	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5		fvopabf4g.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵)
6		fvopabf4g.1	. . 3 ⊢ 𝐶 ∈ V
7	4, 5, 6	fvmpt 6875	. 2 ⊢ (𝐴 ∈ (𝑅 ↑m 𝐷) → (𝐹‘𝐴) = 𝐶)
8	3, 7	syl 17	1 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617

This theorem is referenced by: (None)