Theorem cosnopne 32624

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 6189	. . . . 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 6197	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
7	3, 6	ineq12d 4187	. . 3 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ({𝐴} ∩ {𝐷}))
8		cosnopne.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
9		disjsn2 4679	. . . 4 ⊢ (𝐴 ≠ 𝐷 → ({𝐴} ∩ {𝐷}) = ∅)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐷}) = ∅)
11	7, 10	eqtrd 2765	. 2 ⊢ (𝜑 → (dom {⟨𝐴, 𝐵⟩} ∩ ran {⟨𝐶, 𝐷⟩}) = ∅)
12	11	coemptyd 14952	1 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∩ cin 3916  ∅c0 4299  {csn 4592  ⟨cop 4598  dom cdm 5641  ran crn 5642   ∘ ccom 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653

This theorem is referenced by:  coprprop  32629