Theorem cosnopne 32605

Description: Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)

Hypotheses

Ref	Expression
cosnopne.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
cosnopne.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
cosnopne.1	⊢ (𝜑 → 𝐴 ≠ 𝐷)

Assertion

Ref	Expression
cosnopne	⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)

Proof of Theorem cosnopne

Step	Hyp	Ref	Expression
1		cosnopne.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		dmsnopg 6200	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4		cosnopne.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
5		rnsnopg 6208	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
7	3, 6	ineq12d 4194	. . 3 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ({𝐴} ∩ {𝐷}))
8		cosnopne.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
9		disjsn2 4686	. . . 4 ⊢ (𝐴 ≠ 𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐷}) = ∅)
11	7, 10	eqtrd 2769	. 2 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ∅)
12	11	coemptyd 14987	1 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ≠ wne 2931   ∩ cin 3923  ∅c0 4306  {csn 4599  ⟨cop 4605  dom cdm 5652  ran crn 5653   ∘ ccom 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5118  df-opab 5180  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664

This theorem is referenced by:  coprprop  32610