Theorem tposf1o 50200

Description: Condition of a bijective transposition. (Contributed by Zhi Wang, 5-Oct-2025.)

Assertion

Ref	Expression
tposf1o	⊢ (𝐹:(𝐴 × 𝐵)⟶1-1onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶1-1onto𝐶)

Proof of Theorem tposf1o

Step	Hyp	Ref	Expression
1		relxp 5673	. . 3 ⊢ Rel (𝐴 × 𝐵)
2		tposf1o2 8294	. . 3 ⊢ (Rel (𝐴 × 𝐵) → (𝐹:(𝐴 × 𝐵)⟶1-1onto𝐶 → tpos 𝐹:◡(𝐴 × 𝐵)⟶1-1onto𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶1-1onto𝐶 → tpos 𝐹:◡(𝐴 × 𝐵)⟶1-1onto𝐶)
4		cnvxp 6158	. . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
5		f1oeq2 6832	. . 3 ⊢ (◡(𝐴 × 𝐵) = (𝐵 × 𝐴) → (tpos 𝐹:◡(𝐴 × 𝐵)⟶1-1onto𝐶 ↔ tpos 𝐹:(𝐵 × 𝐴)⟶1-1onto𝐶))
6	4, 5	ax-mp 5	. 2 ⊢ (tpos 𝐹:◡(𝐴 × 𝐵)⟶1-1onto𝐶 ↔ tpos 𝐹:(𝐵 × 𝐴)⟶1-1onto𝐶)
7	3, 6	sylib 221	1 ⊢ (𝐹:(𝐴 × 𝐵)⟶1-1onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶1-1onto𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   × cxp 5653  ^◡ccnv 5654  Rel wrel 5660  ⟶1-1ontowf1o 6554  tpos ctpos 8267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-1st 8022  df-2nd 8023  df-tpos 8268

This theorem is used by:  tposidf1o  50203