Theorem f1o2d 7393

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

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
f1o2d	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

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 f1o2d

Step	Hyp	Ref	Expression
1		f1od.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
2		f1o2d.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
3		f1o2d.3	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
4		f1o2d.4	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))
5	1, 2, 3, 4	f1ocnv2d 7392	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))
6	5	simpld 497	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ↦ cmpt 5138  ^◡ccnv 5548  –1-1-onto→wf1o 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356

This theorem is referenced by:  f1opw2  7394  en3d  8540  f1opwfi  8822  mapfien  8865  djulf1o  9335  djurf1o  9336  fin23lem22  9743  negf1o  11064  incexclem  15185  dvdsflip  15661  hashgcdlem  16119  grplmulf1o  18167  conjghm  18383  gapm  18430  psrbagconf1o  20148  hmeoimaf1o  22372  itg1mulc  24299  resinf1o  25114  eff1olem  25126  sqff1o  25753  dvdsppwf1o  25757  dvdsflf1o  25758  fcobij  30452  frlmsnic  39142