Theorem fcoresf1ob 48142

Description: A composition is bijective iff the restriction of its first component to the minimum domain is bijective and the restriction of its second component to the minimum domain is injective. (Contributed by GL and AV, 7-Oct-2024.)

Hypotheses

Ref	Expression
fcores.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcores.e	⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
fcores.p	⊢ 𝑃 = (◡𝐹 “ 𝐶)
fcores.x	⊢ 𝑋 = (𝐹 ↾ 𝑃)
fcores.g	⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
fcores.y	⊢ 𝑌 = (𝐺 ↾ 𝐸)

Assertion

Ref	Expression
fcoresf1ob	⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))

Proof of Theorem fcoresf1ob

Step	Hyp	Ref	Expression
1		fcores.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		fcores.e	. . . . 5 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
3		fcores.p	. . . . 5 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
4		fcores.x	. . . . 5 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
5		fcores.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
6		fcores.y	. . . . 5 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
7	1, 2, 3, 4, 5, 6	fcoresf1b 48139	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
8	1, 2, 3, 4, 5, 6	fcoresfob 48141	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
9	7, 8	anbi12d 644	. . 3 ⊢ (𝜑 → (((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) ↔ ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) ∧ 𝑌:𝐸–onto→𝐷)))
10		anass 474	. . 3 ⊢ (((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷) ∧ 𝑌:𝐸–onto→𝐷) ↔ (𝑋:𝑃–1-1→𝐸 ∧ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷)))
11	9, 10	bitrdi 290	. 2 ⊢ (𝜑 → (((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) ↔ (𝑋:𝑃–1-1→𝐸 ∧ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷))))
12		df-f1o 6545	. 2 ⊢ ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷))
13		df-f1o 6545	. . 3 ⊢ (𝑌:𝐸–1-1-onto→𝐷 ↔ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷))
14	13	anbi2i 635	. 2 ⊢ ((𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷) ↔ (𝑋:𝑃–1-1→𝐸 ∧ (𝑌:𝐸–1-1→𝐷 ∧ 𝑌:𝐸–onto→𝐷)))
15	11, 12, 14	3bitr4g 317	1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∩ cin 3898  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  ⟶wf 6534  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is used by:  3f1oss1  48144