Theorem tposidres 49994

Description: Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)

Hypotheses

Ref	Expression
tposidres.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
tposidres.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
tposidres	⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

Proof of Theorem tposidres

Step	Hyp	Ref	Expression
1		ovtpos 8258	. . . 4 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (𝑋( I ↾ (𝐴 × 𝐵))𝑌)
2		df-ov 7423	. . . 4 ⊢ (𝑋( I ↾ (𝐴 × 𝐵))𝑌) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
3	1, 2	eqtri 2783	. . 3 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
4		tposidres.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5		tposidres.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	4, 5	opelxpd 5689	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
7	6	fvresd 6905	. . 3 ⊢ (𝜑 → (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩) = ( I ‘⟨𝑋, 𝑌⟩))
8	3, 7	eqtrid 2807	. 2 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ( I ‘⟨𝑋, 𝑌⟩))
9		opex 5431	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
10		fvi 6961	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ V → ( I ‘⟨𝑋, 𝑌⟩) = ⟨𝑋, 𝑌⟩)
11	9, 10	ax-mp 5	. 2 ⊢ ( I ‘⟨𝑋, 𝑌⟩) = ⟨𝑋, 𝑌⟩
12	8, 11	eqtrdi 2811	1 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4589   I cid 5544   × cxp 5648   ↾ cres 5652  ‘cfv 6538  (class class class)co 7420  tpos ctpos 8242

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-ov 7423  df-tpos 8243

This theorem is used by: (None)