Theorem brcogw 4947

Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
brcogw	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → 𝐴(𝐶 ∘ 𝐷)𝐵)

Proof of Theorem brcogw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1031	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → 𝐴 ∈ 𝑉)
2		simpl2 1032	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → 𝐵 ∈ 𝑊)
3		breq2 4132	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴𝐷𝑥 ↔ 𝐴𝐷𝑋))
4		breq1 4131	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝐵 ↔ 𝑋𝐶𝐵))
5	3, 4	anbi12d 477	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)))
6	5	spcegv 2913	. . . 4 ⊢ (𝑋 ∈ 𝑍 → ((𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵) → ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
7	6	imp 124	. . 3 ⊢ ((𝑋 ∈ 𝑍 ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
8	7	3ad2antl3 1192	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
9		brcog 4945	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
10	9	biimpar 297	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) → 𝐴(𝐶 ∘ 𝐷)𝐵)
11	1, 2, 8, 10	syl21anc 1277	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → 𝐴(𝐶 ∘ 𝐷)𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209   class class class wbr 4128   ∘ ccom 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-co 4781

This theorem is referenced by: (None)