Theorem 3f1oss2 48419

Description: The composition of three bijections as bijection from the image of the converse of the domain onto the image of the converse of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)

Assertion

Ref	Expression
3f1oss2	⊢ (((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐻:𝐸⟶1-1onto𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷))

Proof of Theorem 3f1oss2

Step	Hyp	Ref	Expression
1		f1ocnv 6867	. . 3 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → ◡𝐹:𝐵⟶1-1onto𝐴)
2		id 23	. . 3 ⊢ (𝐺:𝐶⟶1-1onto𝐷 → 𝐺:𝐶⟶1-1onto𝐷)
3		f1ocnv 6867	. . 3 ⊢ (𝐻:𝐸⟶1-1onto𝐼 → ◡𝐻:𝐼⟶1-1onto𝐸)
4		3f1oss1 48418	. . 3 ⊢ (((◡𝐹:𝐵⟶1-1onto𝐴 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ ◡𝐻:𝐼⟶1-1onto𝐸) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷))
5	1, 2, 3, 4	syl3anl 1443	. 2 ⊢ (((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐻:𝐸⟶1-1onto𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷))
6		f1orel 6857	. . . . 5 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → Rel 𝐹)
7		dfrel2 6199	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
8	7	biimpi 219	. . . . . . . 8 ⊢ (Rel 𝐹 → ◡◡𝐹 = 𝐹)
9	8	eqcomd 2769	. . . . . . 7 ⊢ (Rel 𝐹 → 𝐹 = ◡◡𝐹)
10	9	coeq2d 5851	. . . . . 6 ⊢ (Rel 𝐹 → ((◡𝐻 ∘ 𝐺) ∘ 𝐹) = ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹))
11	10	f1oeq1d 6849	. . . . 5 ⊢ (Rel 𝐹 → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷)))
12	6, 11	syl 18	. . . 4 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷)))
13	12	3ad2ant1 1151	. . 3 ⊢ ((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐻:𝐸⟶1-1onto𝐼) → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷)))
14	13	adantr 486	. 2 ⊢ (((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐻:𝐸⟶1-1onto𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷)))
15	5, 14	mpbird 260	1 ⊢ (((𝐹:𝐴⟶1-1onto𝐵 ∧ 𝐺:𝐶⟶1-1onto𝐷 ∧ 𝐻:𝐸⟶1-1onto𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)⟶1-1onto(◡𝐻 “ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ⊆ wss 3900  ^◡ccnv 5657   “ cima 5661   ∘ ccom 5662  Rel wrel 5663  ⟶1-1ontowf1o 6565

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 5255  ax-nul 5266  ax-pr 5398

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-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  uspgrlim  49363