Theorem brcogw 5846

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		3simpa 1166	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊))
2		breq2 5107	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐴𝐷𝑥 ↔ 𝐴𝐷𝑋))
3		breq1 5106	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝐵 ↔ 𝑋𝐶𝐵))
4	2, 3	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)))
5	4	spcegv 3552	. . . 4 ⊢ (𝑋 ∈ 𝑍 → ((𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵) → ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
6	5	imp 412	. . 3 ⊢ ((𝑋 ∈ 𝑍 ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
7	6	3ad2antl3 1206	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
8		brcog 5844	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
9	8	biimpar 483	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)) → 𝐴(𝐶 ∘ 𝐷)𝐵)
10	1, 7, 9	syl2an2r 698	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → 𝐴(𝐶 ∘ 𝐷)𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   class class class wbr 5103   ∘ ccom 5655

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-ext 2733  ax-sep 5249  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-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  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-br 5104  df-opab 5168  df-co 5660

This theorem is used by:  utop2nei  24569  utop3cls  24570  iunrelexpuztr  44718  frege96d  44748  frege98d  44752