![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fundcmpsurinj | Structured version Visualization version GIF version |
Description: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.) |
Ref | Expression |
---|---|
fundcmpsurinj | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abrexexg 7958 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V) |
3 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
4 | 3 | sneqd 4636 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → {(𝐹‘𝑥)} = {(𝐹‘𝑦)}) |
5 | 4 | imaeq2d 6057 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ {(𝐹‘𝑦)})) |
6 | 5 | eqeq2d 2738 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)}))) |
7 | 6 | cbvrexvw 3230 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})) |
8 | 7 | abbii 2797 | . . . 4 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} |
9 | 8 | fundcmpsurinjpreimafv 46671 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∧ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))) |
10 | foeq3 6803 | . . . . 5 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → (𝑔:𝐴–onto→𝑝 ↔ 𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})})) | |
11 | f1eq2 6783 | . . . . 5 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → (ℎ:𝑝–1-1→𝐵 ↔ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵)) | |
12 | 10, 11 | 3anbi12d 1434 | . . . 4 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → ((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ (𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∧ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))) |
13 | 12 | 2exbidv 1920 | . . 3 ⊢ (𝑝 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} → (∃𝑔∃ℎ(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ ∃𝑔∃ℎ(𝑔:𝐴–onto→{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∧ ℎ:{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))) |
14 | 2, 9, 13 | spcedv 3583 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑝∃𝑔∃ℎ(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))) |
15 | exrot3 2158 | . 2 ⊢ (∃𝑝∃𝑔∃ℎ(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))) | |
16 | 14, 15 | sylib 217 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2704 ∃wrex 3065 Vcvv 3469 {csn 4624 ◡ccnv 5671 “ cima 5675 ∘ ccom 5676 ⟶wf 6538 –1-1→wf1 6539 –onto→wfo 6540 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |