Theorem f1ocnv2d 6267

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
f1od.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
f1o2d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
f1o2d.3	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
f1o2d.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))

Assertion

Ref	Expression
f1ocnv2d	⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem f1ocnv2d

Step	Hyp	Ref	Expression
1		f1od.1	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
2		f1o2d.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
3		f1o2d.3	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
4		eleq1a 2306	. . . . . 6 ⊢ (𝐶 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵))
5	2, 4	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵))
6	5	impr 379	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → 𝑦 ∈ 𝐵)
7		f1o2d.4	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))
8	7	biimpar 297	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷)
9	8	exp42 371	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑥 = 𝐷))))
10	9	com34 83	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷))))
11	10	imp32 257	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷))
12	6, 11	jcai 311	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))
13		eleq1a 2306	. . . . . 6 ⊢ (𝐷 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴))
14	3, 13	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴))
15	14	impr 379	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → 𝑥 ∈ 𝐴)
16	7	biimpa 296	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶)
17	16	exp42 371	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → 𝑦 = 𝐶))))
18	17	com23 78	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑦 = 𝐶))))
19	18	com34 83	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶))))
20	19	imp32 257	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶))
21	15, 20	jcai 311	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
22	12, 21	impbida 600	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))
23	1, 2, 3, 22	f1ocnvd 6265	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   ↦ cmpt 4176  ^◡ccnv 4753  –1-1-onto→wf1o 5356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364

This theorem is referenced by:  f1o2d  6268  negf1o  8672  negiso  9246  iccf1o  10357  xrnegiso  11972  grpinvcnv  13823  grplactcnv  13857  txhmeo  15310