Theorem tposidres 48864

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 8222	. . . 4 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (𝑋( I ↾ (𝐴 × 𝐵))𝑌)
2		df-ov 7392	. . . 4 ⊢ (𝑋( I ↾ (𝐴 × 𝐵))𝑌) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
3	1, 2	eqtri 2753	. . 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 5679	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
7	6	fvresd 6880	. . 3 ⊢ (𝜑 → (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩) = ( I ‘⟨𝑋, 𝑌⟩))
8	3, 7	eqtrid 2777	. 2 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ( I ‘⟨𝑋, 𝑌⟩))
9		opex 5426	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
10		fvi 6939	. . 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 2781	1 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4597   I cid 5534   × cxp 5638   ↾ cres 5642  ‘cfv 6513  (class class class)co 7389  tpos ctpos 8206

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 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-fv 6521  df-ov 7392  df-tpos 8207

This theorem is referenced by: (None)