Theorem tposidres 49168

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 8183	. . . 4 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (𝑋( I ↾ (𝐴 × 𝐵))𝑌)
2		df-ov 7361	. . . 4 ⊢ (𝑋( I ↾ (𝐴 × 𝐵))𝑌) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
3	1, 2	eqtri 2758	. . 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 5662	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
7	6	fvresd 6853	. . 3 ⊢ (𝜑 → (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩) = ( I ‘⟨𝑋, 𝑌⟩))
8	3, 7	eqtrid 2782	. 2 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ( I ‘⟨𝑋, 𝑌⟩))
9		opex 5411	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
10		fvi 6909	. . 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 2786	1 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3439  ⟨cop 4585   I cid 5517   × cxp 5621   ↾ cres 5625  ‘cfv 6491  (class class class)co 7358  tpos ctpos 8167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-fv 6499  df-ov 7361  df-tpos 8168

This theorem is referenced by: (None)