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Mirrors > Home > MPE Home > Th. List > Mathboxes > fundcmpsurinjlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fundcmpsurinj 46528. (Contributed by AV, 4-Mar-2024.) |
Ref | Expression |
---|---|
fundcmpsurinj.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
fundcmpsurinj.g | ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)})) |
Ref | Expression |
---|---|
fundcmpsurinjlem1 | ⊢ ran 𝐺 = 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fundcmpsurinj.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)})) | |
2 | 1 | rnmpt 5944 | . 2 ⊢ ran 𝐺 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
3 | fundcmpsurinj.p | . 2 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | |
4 | 2, 3 | eqtr4i 2755 | 1 ⊢ ran 𝐺 = 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {cab 2701 ∃wrex 3062 {csn 4620 ↦ cmpt 5221 ◡ccnv 5665 ran crn 5667 “ cima 5669 ‘cfv 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-mpt 5222 df-cnv 5674 df-dm 5676 df-rn 5677 |
This theorem is referenced by: fundcmpsurinjlem2 46518 |
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